Electric potential energy (part 2-- involves calculus)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:05.070
In the last video, we figured
out how much work or how much

00:00:05.070 --> 00:00:07.710
energy we have to put into a
particle to move it within a

00:00:07.710 --> 00:00:08.970
constant electric field.

00:00:08.970 --> 00:00:13.250
Let's see if we can do the same
thing now with a variable

00:00:13.250 --> 00:00:15.340
electric field, and actually,
then we could figure out the

00:00:15.340 --> 00:00:18.070
electric potential of one
position relative to another.

00:00:18.070 --> 00:00:21.220
So let's say we have
a point charge.

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It doesn't have to be a point
charge, but let's just say

00:00:23.580 --> 00:00:30.140
that there's some field being
generated by this charge that

00:00:30.140 --> 00:00:37.320
is a positive q1 coulombs.

00:00:37.320 --> 00:00:39.620
And if we wanted to draw field
lines-- and, of course, this

00:00:39.620 --> 00:00:44.020
is different than the example
with the infinite uniformly

00:00:44.020 --> 00:00:45.900
charged plate because this will
have a variable electric

00:00:45.900 --> 00:00:46.610
field, right?

00:00:46.610 --> 00:00:49.740
What is the electric field
of this charge?

00:00:49.740 --> 00:00:52.512
The electric field is going
to look like this.

00:00:52.512 --> 00:00:54.780
It's essentially the amount of
force it will exert on any

00:00:54.780 --> 00:00:57.350
particle, and it's always going
to be pointed outwards,

00:00:57.350 --> 00:00:59.880
because we assume that the test
particle in question is

00:00:59.880 --> 00:01:02.110
always a positive charge, so
a positive charge will be

00:01:02.110 --> 00:01:04.930
repelled from this
positive charge.

00:01:04.930 --> 00:01:11.980
That field is Coulomb's constant
times this charge q1

00:01:11.980 --> 00:01:15.650
over the distance that we
are from this charge.

00:01:15.650 --> 00:01:19.760
So if I were to draw this
electric field close up, it's

00:01:19.760 --> 00:01:21.010
pretty strong.

00:01:26.640 --> 00:01:30.340
And then as we get further, it
gets weaker a little bit.

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It's always radially outward
from the charge.

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This a bit review for you.

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And this is a vector field that
I'm drawing, where I'm

00:01:41.820 --> 00:01:44.130
just randomly picking points and
showing the vector of the

00:01:44.130 --> 00:01:47.790
field at that point, that it's
pointed outward, and at any

00:01:47.790 --> 00:01:50.370
given radius away from the
circle, these vectors should

00:01:50.370 --> 00:01:51.090
be the same magnitude.

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I know mine aren't exactly,
but hopefully, you get the

00:01:53.040 --> 00:01:54.310
point, right?

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And the magnitude of these
vectors decrease with the

00:01:57.860 --> 00:02:00.830
square of the distance.

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So that's what the field is
going to look like if I were

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to draw a bunch of vector
field lines.

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And that makes sense, just
as a review, right?

00:02:06.240 --> 00:02:09.979
Because we know the force-- if
we had another test charge q2,

00:02:09.979 --> 00:02:12.600
we know that the force on that
test charge is just q2 times

00:02:12.600 --> 00:02:15.400
this, and that's just Coulomb's
Law, right?

00:02:15.400 --> 00:02:20.450
That the force on some other
particle q2 is equal to the

00:02:20.450 --> 00:02:30.560
electric field times q2, which
is equal to k q1 q2 over r

00:02:30.560 --> 00:02:31.060
squared, right?

00:02:31.060 --> 00:02:34.180
This is just Coulomb's Law, and
actually this comes from

00:02:34.180 --> 00:02:35.520
Coulomb's Law.

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So since we know that, let's
take some other positive

00:02:40.880 --> 00:02:43.190
particle out here, and
let's call that q2.

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I'll write it up here since I
already messed up down there.

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q2, and let's say this is a
positive charge, so it is

00:02:53.260 --> 00:02:59.540
going to be repelled
from this q1.

00:02:59.540 --> 00:03:04.100
And let's figure out how much
work does it take to push in

00:03:04.100 --> 00:03:05.870
this particle a certain
distance, right?

00:03:05.870 --> 00:03:08.610
Because the field is
pushing it outward.

00:03:08.610 --> 00:03:11.180
It takes work to
push it inward.

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So let's say we want
to push it in.

00:03:13.350 --> 00:03:15.200
Let's say it's at 10 meters.

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Let's say that this distance
right here-- let me draw a

00:03:18.810 --> 00:03:26.260
radial line-- let's say that
this distance right here is 10

00:03:26.260 --> 00:03:33.720
meters, and I want to push this
particle in 5 meters, so

00:03:33.720 --> 00:03:37.320
it eventually gets right here.

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This is where I'm eventually
going to get it so then it's

00:03:38.970 --> 00:03:40.260
going to be 5 meters away.

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So how much work does it take
to move it 5 meters towards

00:03:47.430 --> 00:03:48.730
this charge?

00:03:48.730 --> 00:03:52.070
Well, the way you think about
it is the field keeps

00:03:52.070 --> 00:03:53.230
changing, right?

00:03:53.230 --> 00:03:56.880
But we can assume over a very,
very, very, very infinitely

00:03:56.880 --> 00:04:00.240
small distance, and let's call
that infinitely small distance

00:04:00.240 --> 00:04:04.660
dr, change in radius, and as
you can see, we're about to

00:04:04.660 --> 00:04:08.500
embark on some integral and
differential calculus.

00:04:08.500 --> 00:04:10.720
If you don't understand what any
of this is, you might want

00:04:10.720 --> 00:04:14.250
to review or learn the calculus
in the calculus

00:04:14.250 --> 00:04:18.040
playlist, but how much work does
it require to move this

00:04:18.040 --> 00:04:20.019
particle a very, very
small distance?

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Well, let's just assume over
this very, very, very small

00:04:22.720 --> 00:04:25.940
distance, that the electric
field is roughly constant, and

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so we can say that the very,
very small amount of work to

00:04:29.600 --> 00:04:32.750
move over that very, very small
distance is equal to

00:04:32.750 --> 00:04:40.820
Coulomb's constant q1 q2 over
r squared times dr.

00:04:40.820 --> 00:04:42.150
Now before we move on,
let's think about

00:04:42.150 --> 00:04:43.400
something for a second.

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Coulomb's Law tells us that this
is the outward force that

00:04:50.290 --> 00:04:53.140
this charge is exerting on
this particle or that the

00:04:53.140 --> 00:04:55.570
field is exerting on
this particle.

00:04:55.570 --> 00:04:59.330
The force that we have to apply
to move the particle

00:04:59.330 --> 00:05:02.380
from here to here has to
be an inward force.

00:05:02.380 --> 00:05:06.160
It has to be in the exact
opposite direction, so it has

00:05:06.160 --> 00:05:07.410
to be a negative.

00:05:07.410 --> 00:05:08.300
And why is that?

00:05:08.300 --> 00:05:12.230
Because we have to completely
offset the force of the field.

00:05:12.230 --> 00:05:14.630
Maybe if the particle was
already moving a little bit,

00:05:14.630 --> 00:05:17.280
then our force will keep it
from decelerating from the

00:05:17.280 --> 00:05:19.760
field, and if it wasn't already
moving, we would have

00:05:19.760 --> 00:05:22.910
to nudge it just an infinitely
small amount just to get it

00:05:22.910 --> 00:05:26.010
moving, and then our force would
completely offset the

00:05:26.010 --> 00:05:27.970
force of the field, and the
particle would neither

00:05:27.970 --> 00:05:30.520
accelerate nor decelerate.

00:05:30.520 --> 00:05:32.890
So this is the amount of work,
and I just want to explain

00:05:32.890 --> 00:05:34.660
that we want to put that
negative sign there because we

00:05:34.660 --> 00:05:37.350
going in the opposite direction
of the field.

00:05:37.350 --> 00:05:40.210
So how do we figure out the
total amount of work?

00:05:40.210 --> 00:05:43.000
We figured out the amount of
work to get it from here to

00:05:43.000 --> 00:05:46.140
here, and I even drew it much
bigger than it would be.

00:05:46.140 --> 00:05:48.000
These dr's, this is
an infinitely

00:05:48.000 --> 00:05:51.010
small change in radius.

00:05:51.010 --> 00:05:52.890
If we want to figure out the
total work, then we keep

00:05:52.890 --> 00:05:53.450
adding them up.

00:05:53.450 --> 00:05:55.770
We say, OK, what's the work to
go from here to here, then the

00:05:55.770 --> 00:05:57.480
work to go from there to there,
then the work to go

00:05:57.480 --> 00:06:01.550
from there to there, all the way
until we get to 5 meters

00:06:01.550 --> 00:06:04.190
away from this charge.

00:06:04.190 --> 00:06:07.740
And what we do when we take the
sum of these, we assume

00:06:07.740 --> 00:06:11.190
that it's an infinite sum of
infinitely small increments.

00:06:11.190 --> 00:06:15.250
And as you learned, that is
nothing but the integral, and

00:06:15.250 --> 00:06:23.790
so that is the total work is
equal to the integral.

00:06:23.790 --> 00:06:25.260
That's going to be a definite
integral because we're

00:06:25.260 --> 00:06:26.210
starting at this point.

00:06:26.210 --> 00:06:29.900
We're summing from-- our radius
is equal to 10 meters--

00:06:29.900 --> 00:06:33.800
that's our starting point--
to radius equals 5 meters.

00:06:33.800 --> 00:06:36.270
That might be a little
unintuitive that we're

00:06:36.270 --> 00:06:39.020
starting at the higher value and
ending at the lower value,

00:06:39.020 --> 00:06:39.640
but that's what we're doing.

00:06:39.640 --> 00:06:41.780
We're pushing it inwards.

00:06:41.780 --> 00:06:50.210
And then we're taking the
integral of minus k q1 q2 over

00:06:50.210 --> 00:06:55.550
r squared dr. All of these are
constant terms up here, right?

00:06:55.550 --> 00:06:58.030
So we could take them out.

00:06:58.030 --> 00:07:01.520
So this is the same thing--
I don't want to run out of

00:07:01.520 --> 00:07:14.350
space-- as minus k q1 q2 times
the integral from 10 to 5 of 1

00:07:14.350 --> 00:07:20.950
over r squared-- or to the
negative 2-- dr. And that

00:07:20.950 --> 00:07:28.310
equals minus k-- I'm running
out of space-- q1 q2.

00:07:28.310 --> 00:07:29.140
We take the antiderivative.

00:07:29.140 --> 00:07:30.720
We don't have to worry about
plus here because it's a

00:07:30.720 --> 00:07:31.600
definite integral.

00:07:31.600 --> 00:07:33.580
r to the negative 2, what's
the antiderivative?

00:07:33.580 --> 00:07:35.890
It's minus r to the
negative 1.

00:07:35.890 --> 00:07:40.900
Well, that minus r, the minus
on the minus r will just

00:07:40.900 --> 00:07:41.760
cancel with this.

00:07:41.760 --> 00:07:46.340
That becomes a plus r to the
negative 1, And you evaluate

00:07:46.340 --> 00:07:50.130
it at 5 and then subtract it
and evaluate it at 10.

00:07:50.130 --> 00:07:52.370
And then-- let me
just go up here.

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Actually, let me erase
some of this.

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Let me erase this up here.

00:08:04.350 --> 00:08:06.040
Valuable space to work on.

00:08:08.840 --> 00:08:12.010
So we said that the work
is equal to-- I'll

00:08:12.010 --> 00:08:13.100
just rewrite that.

00:08:13.100 --> 00:08:15.040
We had the minus out here, but
then we had a minus and we

00:08:15.040 --> 00:08:17.080
took the antiderivative and they
canceled out, so we have

00:08:17.080 --> 00:08:25.990
k q1 q2 times the antiderivative
evaluated at 5,

00:08:25.990 --> 00:08:27.780
so 1 over 5, right?

00:08:27.780 --> 00:08:30.960
r to the negative 1, so 1 over
r minus the antiderivative

00:08:30.960 --> 00:08:36.250
evaluated at 10 minus 1 over 10
is equal to-- well, 1 over

00:08:36.250 --> 00:08:41.049
5, that's the same thing
as 2 over 10, right?

00:08:41.049 --> 00:08:50.280
So we have the work is equal
to k q1 q2 times 2 over 10

00:08:50.280 --> 00:08:55.360
minus 1 over 10 is 1 over
10, so that equals

00:08:55.360 --> 00:09:00.740
k q1 q2 over 10.

00:09:00.740 --> 00:09:04.760
That's the work to take the
particle from here to here.

00:09:04.760 --> 00:09:07.740
And so similarly, we could say
that the potential energy of

00:09:07.740 --> 00:09:11.120
the particle at this-- the
potential difference of the

00:09:11.120 --> 00:09:15.820
particle at this point relative
to this point, that

00:09:15.820 --> 00:09:19.610
the potential difference here
is this much higher.

00:09:19.610 --> 00:09:21.670
It's going to be in joules
because that's the unit of

00:09:21.670 --> 00:09:24.900
energy or work or potential,
because

00:09:24.900 --> 00:09:25.940
potential is energy anyway.

00:09:25.940 --> 00:09:28.960
So the electric potential
difference between this point

00:09:28.960 --> 00:09:34.030
and this point: at this point,
it is this value higher.

00:09:34.030 --> 00:09:38.510
Let's do another example, and
actually, you might just find

00:09:38.510 --> 00:09:41.790
this interesting, just from
something to think about.

00:09:41.790 --> 00:09:45.550
The big difference between--
actually, let me just continue

00:09:45.550 --> 00:09:47.510
this into the next video because
I realize I'm already

00:09:47.510 --> 00:09:48.195
at 10 minutes.

00:09:48.195 --> 00:09:49.880
I'll see you soon.


Electric potential energy

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.640
Let's review a little bit of
what we had learned many, many

00:00:03.640 --> 00:00:06.640
videos ago about gravitational
potential energy and then see

00:00:06.640 --> 00:00:10.005
if we can draw the analogy,
which is actually very strong,

00:00:10.005 --> 00:00:12.640
to electrical potential
energy.

00:00:12.640 --> 00:00:13.380
So what do we know about

00:00:13.380 --> 00:00:14.660
gravitational potential energy?

00:00:14.660 --> 00:00:17.930
If we said this was the surface
of the Earth-- we

00:00:17.930 --> 00:00:21.390
don't have to be on Earth, but
it makes visualization easy.

00:00:21.390 --> 00:00:24.460
We could be anywhere that has
gravity, and the potential

00:00:24.460 --> 00:00:27.850
energy would be due to the
gravitational field of that

00:00:27.850 --> 00:00:29.800
particular mass, but let's
say this is the

00:00:29.800 --> 00:00:31.500
surface of the Earth.

00:00:31.500 --> 00:00:38.270
We learned that if we have some
mass m up here and that

00:00:38.270 --> 00:00:42.610
the gravitational field at this
area-- or at least the

00:00:42.610 --> 00:00:45.770
gravitational acceleration-- is
g, or 9.8 meters per second

00:00:45.770 --> 00:00:49.775
squared, and it is h-- we could
say, I guess, meters,

00:00:49.775 --> 00:00:52.070
but we could use any units.

00:00:52.070 --> 00:00:54.690
Let's say it is h meters above
the ground, that the

00:00:54.690 --> 00:00:59.900
gravitational potential energy
of this object at that point

00:00:59.900 --> 00:01:05.340
is equal to the mass times the
acceleration of gravity times

00:01:05.340 --> 00:01:11.870
the height, or you could view
it as the force of gravity,

00:01:11.870 --> 00:01:13.550
the magnitude of the
force of gravity.

00:01:13.550 --> 00:01:15.172
You know, it's a vector, but
we can say the magnitude of

00:01:15.172 --> 00:01:19.790
the vector times height.

00:01:19.790 --> 00:01:21.250
And so what is potential
energy?

00:01:21.250 --> 00:01:23.010
Well, we know that if something
has potential energy

00:01:23.010 --> 00:01:25.740
and if nothing is stopping it
and we just let go, that

00:01:25.740 --> 00:01:28.580
energy, at least with
gravitational potential

00:01:28.580 --> 00:01:31.840
energy, the object will start
accelerating downwards, and a

00:01:31.840 --> 00:01:34.350
lot of that potential energy,
and eventually all of it, will

00:01:34.350 --> 00:01:36.990
be converted to kinetic
energy.

00:01:36.990 --> 00:01:40.730
So potential energy is energy
that is being stored by an

00:01:40.730 --> 00:01:44.180
object's situation or kind of
this notional energy that an

00:01:44.180 --> 00:01:46.530
object has by virtue
of where it is.

00:01:46.530 --> 00:01:50.550
So in order for something to
have this notional energy,

00:01:50.550 --> 00:01:53.200
some energy must have
been put into it.

00:01:53.200 --> 00:01:55.700
And as we learned with
gravitational potential

00:01:55.700 --> 00:01:58.600
energy, you could view
gravitational potential energy

00:01:58.600 --> 00:02:03.780
as the work necessary to move
an object to that position.

00:02:03.780 --> 00:02:06.840
Now, if we're talking about work
to move something into

00:02:06.840 --> 00:02:09.050
that position, or whatever, we
always have to think about,

00:02:09.050 --> 00:02:11.000
well, move it from where?

00:02:11.000 --> 00:02:12.720
Well, when we talk about
gravitational potential

00:02:12.720 --> 00:02:15.610
energy, we're talking about
moving it from the surface of

00:02:15.610 --> 00:02:18.990
the Earth, right?

00:02:18.990 --> 00:02:22.180
And so how much work is required
to move that same

00:02:22.180 --> 00:02:25.040
mass-- let's say it was here at
first-- to move it from a

00:02:25.040 --> 00:02:28.140
height of zero to
a height of h?

00:02:28.140 --> 00:02:30.950
Well, the whole time, the
Earth, or the force of

00:02:30.950 --> 00:02:35.940
gravity, is going to
be F sub g, right?

00:02:35.940 --> 00:02:39.890
So essentially, if I'm pulling
it or pushing it upwards, I'm

00:02:39.890 --> 00:02:41.670
going to have to have-- and
let's say at a constant

00:02:41.670 --> 00:02:45.130
velocity-- I'm going to have to
have an equal and opposite

00:02:45.130 --> 00:02:48.940
force to its weight
to pull it up.

00:02:48.940 --> 00:02:50.240
Otherwise, it would accelerate
downwards.

00:02:50.240 --> 00:02:52.870
I'd have to do a little bit more
just to get it moving, to

00:02:52.870 --> 00:02:55.510
accelerate it however much, but
then once I get it just

00:02:55.510 --> 00:02:58.510
accelerating, essentially I
would have to apply an upward

00:02:58.510 --> 00:03:01.830
force, which is equivalent to
the downward force of gravity,

00:03:01.830 --> 00:03:05.110
and I would do it for a
distance of h, right?

00:03:05.110 --> 00:03:05.720
What is work?

00:03:05.720 --> 00:03:08.850
Work is just force
times distance.

00:03:08.850 --> 00:03:11.850
Force times distance, and it
has to be force in the

00:03:11.850 --> 00:03:13.860
direction of the distance.

00:03:13.860 --> 00:03:17.650
So what's the work necessary
to get this mass up here?

00:03:17.650 --> 00:03:25.370
Well, the work is equal to the
force of gravity times height,

00:03:25.370 --> 00:03:26.620
so it's equal to the

00:03:26.620 --> 00:03:28.870
gravitational potential energy.

00:03:28.870 --> 00:03:30.110
Now this is an interesting
thing.

00:03:30.110 --> 00:03:34.550
Notice we picked the reference
point as the surface of the

00:03:34.550 --> 00:03:36.470
Earth, but we could
have picked any

00:03:36.470 --> 00:03:37.620
arbitrary reference point.

00:03:37.620 --> 00:03:40.900
We could have said, well, from
10 meters below the surface of

00:03:40.900 --> 00:03:43.170
the Earth, which could have been
down here, or we could

00:03:43.170 --> 00:03:45.410
have actually said, you know,
from a platform that's 5

00:03:45.410 --> 00:03:47.156
meters above the Earth.

00:03:47.156 --> 00:03:50.520
So it actually turns out, when
you think of it that way, that

00:03:50.520 --> 00:03:53.980
potential energy of any form,
but especially gravitational

00:03:53.980 --> 00:03:56.400
potential energy-- and we'll
see electrical potential

00:03:56.400 --> 00:04:00.430
energy-- it's always in
reference to some other point,

00:04:00.430 --> 00:04:03.800
so it's really a change in
potential energy that matters.

00:04:03.800 --> 00:04:05.490
And I know when we studied
potential energy, it seemed

00:04:05.490 --> 00:04:07.620
like there was kind of an
absolute potential energy, but

00:04:07.620 --> 00:04:09.720
that's because we always assume
that the potential

00:04:09.720 --> 00:04:12.710
energy of something is zero the
surface of the Earth and

00:04:12.710 --> 00:04:15.210
that we want to know the
potential energy relative to

00:04:15.210 --> 00:04:17.459
the surface of the Earth, so it
would be kind of, you know,

00:04:17.459 --> 00:04:19.440
how much work does it take to
take something from the

00:04:19.440 --> 00:04:21.440
surface of the Earth
to that height?

00:04:21.440 --> 00:04:24.710
But really, we should be saying,
well, the potential

00:04:24.710 --> 00:04:28.320
energy of gravity-- like this
statement shouldn't be, you

00:04:28.320 --> 00:04:29.480
know, this is just the absolute

00:04:29.480 --> 00:04:30.890
potential energy of gravity.

00:04:30.890 --> 00:04:33.330
We should say this is the
potential energy of gravity

00:04:33.330 --> 00:04:39.230
relative to the surface of the
Earth is equal to the work

00:04:39.230 --> 00:04:44.420
necessary to move something, to
move that same mass, from

00:04:44.420 --> 00:04:47.780
the surface of the Earth to
its current position.

00:04:47.780 --> 00:04:53.630
We could have defined some other
term that is not really

00:04:53.630 --> 00:04:58.520
used, but we could have said
potential energy of gravity

00:04:58.520 --> 00:05:02.310
relative to minus 5 meters
below the surface of the

00:05:02.310 --> 00:05:04.080
Earth, and that would be the
work necessary to move

00:05:04.080 --> 00:05:08.070
something from minus 5 meters
to its current height.

00:05:08.070 --> 00:05:09.130
And, of course, that
might matter.

00:05:09.130 --> 00:05:11.500
What if we cut up a hole and
we want to see what is the

00:05:11.500 --> 00:05:12.350
kinetic energy here?

00:05:12.350 --> 00:05:14.520
Well, then that potential
energy would matter.

00:05:14.520 --> 00:05:18.170
Anyway, so I just wanted to do
this review of potential

00:05:18.170 --> 00:05:22.960
energy because now it'll make
the jump to electrical

00:05:22.960 --> 00:05:24.890
potential energy all that
easier, because you'll

00:05:24.890 --> 00:05:27.380
actually see it's pretty
much the same thing.

00:05:27.380 --> 00:05:30.170
It's just the source of the
field and the source of the

00:05:30.170 --> 00:05:31.630
potential is something
different.

00:05:36.570 --> 00:05:39.550
So electrical potential energy,
just actually we know

00:05:39.550 --> 00:05:43.270
that gravitational fields are
not constant, we can assume

00:05:43.270 --> 00:05:45.140
they're constant maybe near the
surface of the Earth and

00:05:45.140 --> 00:05:47.770
all that, but we also know that
electrical fields aren't

00:05:47.770 --> 00:05:50.180
constant, and actually they
have very similar formulas.

00:05:50.180 --> 00:05:53.720
But just for the simplicity of
explaining it, let's assume a

00:05:53.720 --> 00:05:54.900
constant electric field.

00:05:54.900 --> 00:05:57.190
And if you don't believe me that
one can be constructed,

00:05:57.190 --> 00:06:00.620
you should watch my videos that
involve a reasonable bit

00:06:00.620 --> 00:06:05.930
of calculus that show that a
uniform electric field can be

00:06:05.930 --> 00:06:12.520
generated by an infinite
uniformly charged plane.

00:06:12.520 --> 00:06:18.210
Let's say this is the side view
of an infinite uniformly

00:06:18.210 --> 00:06:20.960
charged plane and let's
say that this

00:06:20.960 --> 00:06:22.700
is positively charged.

00:06:25.480 --> 00:06:27.490
Of course, you can never get
a proper side view of an

00:06:27.490 --> 00:06:29.370
infinite plane, because you
can never kind of cut it,

00:06:29.370 --> 00:06:32.390
because it's infinite in every
direction, but let's say that

00:06:32.390 --> 00:06:35.890
this one is and this
is the side view.

00:06:35.890 --> 00:06:38.560
So first of all, let's think
about its electric field.

00:06:38.560 --> 00:06:41.230
It's electric field is going to
point upward, and how do we

00:06:41.230 --> 00:06:43.740
know it points upward?

00:06:43.740 --> 00:06:46.450
Because the electric field is
essentially what is-- and this

00:06:46.450 --> 00:06:47.530
is just a convention.

00:06:47.530 --> 00:06:51.330
What would a positive charge
do in the field?

00:06:51.330 --> 00:06:54.090
Well, if this plate is positive,
a positive charge,

00:06:54.090 --> 00:06:55.530
we're going to want to
get away from it.

00:06:55.530 --> 00:06:58.440
So we know the electric field
points upward and we know that

00:06:58.440 --> 00:07:01.360
it's constant, that if these
were field vectors, that

00:07:01.360 --> 00:07:03.510
they're going to be the same
size, no matter how far away

00:07:03.510 --> 00:07:07.240
we get from the source
of the field.

00:07:07.240 --> 00:07:09.310
And I'm just going to pick
a number for the

00:07:09.310 --> 00:07:10.050
strength of the field.

00:07:10.050 --> 00:07:15.050
We actually proved in those
fancy videos that I made on

00:07:15.050 --> 00:07:18.340
the uniform electric field of an
infinite, uniformly charged

00:07:18.340 --> 00:07:21.010
plane that we actually proved
how you could calculate it.

00:07:21.010 --> 00:07:28.130
But let's just say that this
electric field is equal to 5

00:07:28.130 --> 00:07:29.060
newtons per coulomb.

00:07:29.060 --> 00:07:32.700
That's actually quite strong,
but it makes the math easy.

00:07:32.700 --> 00:07:38.020
So my question to you is how
much work does it take to take

00:07:38.020 --> 00:07:42.280
a positive point charge-- let
me pick a different color.

00:07:45.340 --> 00:07:48.200
Let's say this is the
starting position.

00:07:48.200 --> 00:07:52.770
It's a positive 2 coulombs.

00:07:52.770 --> 00:07:55.440
Once again, that's a massive
point charge, but

00:07:55.440 --> 00:07:56.700
we want easy numbers.

00:07:56.700 --> 00:08:04.950
How much work does it take it to
move that 2-coulomb charge

00:08:04.950 --> 00:08:09.240
3 meters within this field?

00:08:09.240 --> 00:08:09.830
How much work?

00:08:09.830 --> 00:08:12.510
So we're going to start here
and we're going to move it

00:08:12.510 --> 00:08:15.110
down towards the plate 3
meters, and it's ending

00:08:15.110 --> 00:08:18.120
position is going to be
right here, right?

00:08:18.120 --> 00:08:19.300
That's when it's done.

00:08:19.300 --> 00:08:21.790
How much work does that take?

00:08:21.790 --> 00:08:25.600
Well, what is the force of
the field right here?

00:08:25.600 --> 00:08:30.430
What is the force exerted on
this 2-coulomb charge?

00:08:30.430 --> 00:08:34.840
Well, electric field is just
force per charge, right?

00:08:34.840 --> 00:08:37.950
So if you want to know the force
of the field at that

00:08:37.950 --> 00:08:41.570
point-- let me draw that
in a different color.

00:08:41.570 --> 00:08:46.090
The force of the field acting on
it, so let's say the field

00:08:46.090 --> 00:08:51.010
force, or the force of the
field, actually, is going to

00:08:51.010 --> 00:08:57.040
be equal to 5 newtons per
coulomb times 2 coulombs,

00:08:57.040 --> 00:08:58.590
which is equal to 10 newtons.

00:08:58.590 --> 00:09:00.160
We know it's going to be upward,
because this is a

00:09:00.160 --> 00:09:04.090
positive charge, and this is a
positively charged infinite

00:09:04.090 --> 00:09:07.960
plate, so we know this is an
upward force of 10 newtons.

00:09:07.960 --> 00:09:12.190
So in order to get this charge,
to pull it down or to

00:09:12.190 --> 00:09:14.710
push it down here, we
essentially have to exert a

00:09:14.710 --> 00:09:17.690
force of 10 newtons
downwards, right?

00:09:17.690 --> 00:09:20.800
Exert a force of 10 newtons in
the direction of the movement.

00:09:20.800 --> 00:09:23.610
And, of course, just like we did
with gravity, we have to

00:09:23.610 --> 00:09:25.820
maybe do a little bit more than
that just to accelerate

00:09:25.820 --> 00:09:27.985
it a little bit just so you have
some net downward force,

00:09:27.985 --> 00:09:31.020
but once you do, you just have
to completely balance the

00:09:31.020 --> 00:09:31.920
upward force.

00:09:31.920 --> 00:09:34.930
So just for our purposes, you
have a 10-newton force

00:09:34.930 --> 00:09:38.110
downward and you apply that
force for a distance of 3

00:09:38.110 --> 00:09:41.870
meters, the work that you put to
take this 2-coulomb charge

00:09:41.870 --> 00:09:49.050
from here to here, the work
is going to be equal to 10

00:09:49.050 --> 00:09:53.850
newtons-- that's the force--
times 3 meters.

00:09:53.850 --> 00:09:58.350
So the work is going to equal
30 newton-meters, which is

00:09:58.350 --> 00:09:59.920
equal to 30 joules.

00:09:59.920 --> 00:10:02.540
A joule is just a
newton-meter.

00:10:02.540 --> 00:10:07.000
And so we can now say since it
took us 30 joules of energy to

00:10:07.000 --> 00:10:11.810
move this charge from here to
here, that within this uniform

00:10:11.810 --> 00:10:18.650
electric field, the potential
energy of the charge here is

00:10:18.650 --> 00:10:21.880
relative to the charge here.

00:10:21.880 --> 00:10:25.400
You always have to pick a point
relative to where the

00:10:25.400 --> 00:10:28.600
potential is, so the electrical
potential energy

00:10:28.600 --> 00:10:35.310
here relative to here and this
is electrical potential

00:10:35.310 --> 00:10:41.690
energy, and you could say P2
relative to P1-- I'm using my

00:10:41.690 --> 00:10:44.140
made-up notation, but that gives
you a sense of what it

00:10:44.140 --> 00:10:50.830
is-- is equal to 30 joules.

00:10:50.830 --> 00:10:51.870
And how could that help us?

00:10:51.870 --> 00:10:54.680
Well, if we also knew the mass--
let's say that this

00:10:54.680 --> 00:10:57.930
charge had some mass.

00:10:57.930 --> 00:11:02.690
We would know that if we let go
of this object, by the time

00:11:02.690 --> 00:11:06.000
it got here, that 30 joules
would be-- essentially

00:11:06.000 --> 00:11:08.600
assuming that none of it got
transmitted to heat or

00:11:08.600 --> 00:11:11.180
resistance or whatever-- we know
that all of it would be

00:11:11.180 --> 00:11:13.200
kinetic energy at this point.

00:11:13.200 --> 00:11:14.870
So actually, we could
work it out.

00:11:14.870 --> 00:11:24.090
Let's say that this does have
a mass of 1 kilogram and we

00:11:24.090 --> 00:11:27.210
were to just let go
of it, right?

00:11:27.210 --> 00:11:32.330
We used some force to bring it
down here, and then we let go.

00:11:32.330 --> 00:11:34.755
So we know that the electric
field is going to accelerate

00:11:34.755 --> 00:11:36.320
it upwards, right?

00:11:36.320 --> 00:11:39.160
It's going to exert an upward
force of 5 newtons per

00:11:39.160 --> 00:11:42.990
coulomb, and the thing's going
to keep [COUGHS]--

00:11:42.990 --> 00:11:44.920
excuse me-- keep accelerating
until it gets

00:11:44.920 --> 00:11:47.260
to this point, right?

00:11:47.260 --> 00:11:49.290
What's its velocity going
to be at that point?

00:11:49.290 --> 00:11:51.610
Well, all of this electrical
potential energy is going to

00:11:51.610 --> 00:11:53.610
be converted to kinetic
energy.

00:11:53.610 --> 00:11:57.970
So essentially, we have 30
joules is going to be equal to

00:11:57.970 --> 00:12:01.040
1/2 mv squared, right?

00:12:01.040 --> 00:12:05.450
We know the mass, I said, is 1,
so we get 60 is equal to v

00:12:05.450 --> 00:12:09.250
squared, so the velocity is the
square root of 60, so it's

00:12:09.250 --> 00:12:13.930
7 point something, something,
something meters per second.

00:12:13.930 --> 00:12:16.930
So if I just pull that charge
down, and it has a mass of 1

00:12:16.930 --> 00:12:20.070
kilogram, and I let go, it's
just going to accelerate and

00:12:20.070 --> 00:12:22.120
be going pretty fast once
it gets to this point.

00:12:22.120 --> 00:12:24.785
Anyway, I'm 12 minutes into this
video, so I will continue

00:12:24.785 --> 00:12:27.270
in the next, but hopefully, that
gives you a sense of what

00:12:27.270 --> 00:12:30.990
electrical potential energy
is, and really, it's no

00:12:30.990 --> 00:12:32.970
different than gravitational
potential energy.

00:12:32.970 --> 00:12:35.080
It's just the source of the
field is different.

00:12:35.080 --> 00:12:36.650
See you soon.


Renting vs. Buying (detailed analysis)

WEBVTT
Kind: captions
Language: en

00:00:01.080 --> 00:00:01.990
Welcome.

00:00:01.990 --> 00:00:05.930
So I've done this series of
presentations about housing.

00:00:05.930 --> 00:00:09.670
And at least, my thesis on why
housing prices might have gone

00:00:09.670 --> 00:00:12.260
up, and how you should maybe,
in simple terms, think about

00:00:12.260 --> 00:00:14.220
the rent-versus-buy decision.

00:00:14.220 --> 00:00:16.420
But one thing that's happened, a
lot of people said, oh, Sal,

00:00:16.420 --> 00:00:19.200
you're making oversimplifying
assumptions.

00:00:19.200 --> 00:00:22.870
You're assuming interest-only
loans.

00:00:22.870 --> 00:00:25.490
You're not factoring in the tax
deductions of mortgages,

00:00:25.490 --> 00:00:27.300
et cetera, of interest
on your mortgage.

00:00:27.300 --> 00:00:29.850
Which I did, but I did make some
simplifying assumptions.

00:00:29.850 --> 00:00:32.900
So that we could kind of do back
of the envelope math, and

00:00:32.900 --> 00:00:34.670
just think about what the main
drivers are when you think

00:00:34.670 --> 00:00:36.210
about renting versus buying.

00:00:36.210 --> 00:00:38.810
But it is fair.

00:00:38.810 --> 00:00:39.930
That's just kind
of a first cut.

00:00:39.930 --> 00:00:43.950
You really should do a
multi-line model, trying to

00:00:43.950 --> 00:00:45.920
figure out what could
happen to you.

00:00:45.920 --> 00:00:47.460
And then tweak your
assumptions.

00:00:47.460 --> 00:00:50.080
And really figure out what's
going to happen to you if

00:00:50.080 --> 00:00:51.770
housing appreciates,
depreciates.

00:00:51.770 --> 00:00:53.170
If interest rates change.

00:00:53.170 --> 00:00:57.850
If you put 10% down, or
20% down, or whatever.

00:00:57.850 --> 00:01:00.250
So with that in mind, I've
constructed this model.

00:01:00.250 --> 00:01:03.350
What I called, this is the
home purchase model.

00:01:03.350 --> 00:01:05.570
And you can download it yourself
and play with it.

00:01:05.570 --> 00:01:08.390
I think this will prove
to be useful for you.

00:01:08.390 --> 00:01:10.765
You can download it at
khanacademy.org/

00:01:10.765 --> 00:01:12.015
downloads/buyrent.xls.

00:01:15.930 --> 00:01:17.280
It's an Excel spreadsheet.

00:01:17.280 --> 00:01:19.060
So if you have Excel, you should
be able to access it.

00:01:19.060 --> 00:01:20.480
And maybe you want to
follow along while

00:01:20.480 --> 00:01:21.730
you watch this video.

00:01:21.730 --> 00:01:22.980
So just khanacademy.org/
downloads/buyrent.xls.

00:01:33.630 --> 00:01:35.860
So once you download it,
let me explain what I

00:01:35.860 --> 00:01:38.390
assumed in the model.

00:01:38.390 --> 00:01:43.030
So what I did in yellow, both
this bright yellow and this

00:01:43.030 --> 00:01:45.330
less bright yellow, these
are our assumptions.

00:01:45.330 --> 00:01:48.090
These are the things that are
going to drive the model, and

00:01:48.090 --> 00:01:53.050
tell us whether over -- and I
calculated over 10 years --

00:01:53.050 --> 00:01:57.630
whether we will do better
renting versus buying.

00:01:57.630 --> 00:02:00.120
And so if you download this
model and want to play with it

00:02:00.120 --> 00:02:03.470
yourself, unless you are fairly
sophisticated with

00:02:03.470 --> 00:02:06.250
Excel, the only things
you should change are

00:02:06.250 --> 00:02:07.270
the things in yellow.

00:02:07.270 --> 00:02:08.639
Everything else is calculated.

00:02:08.639 --> 00:02:10.509
And it's driven by
these inputs.

00:02:10.509 --> 00:02:11.660
So of course, what matters
in a home?

00:02:11.660 --> 00:02:13.540
Well the purchase
price matters.

00:02:13.540 --> 00:02:14.810
So you just in put it there.

00:02:14.810 --> 00:02:16.250
The down payment matters.

00:02:16.250 --> 00:02:18.480
You could, if you want, you can
just write like I wrote,

00:02:18.480 --> 00:02:20.560
20% of whatever the
purchase price is.

00:02:20.560 --> 00:02:23.370
So you can write the exact
number, or you can just leave

00:02:23.370 --> 00:02:23.995
it the way I did.

00:02:23.995 --> 00:02:26.370
And whatever the down payment
percentage is,

00:02:26.370 --> 00:02:27.690
it'll calculate it.

00:02:27.690 --> 00:02:29.510
This is the interest
rate you assume.

00:02:29.510 --> 00:02:31.140
This is the principal
amortization.

00:02:31.140 --> 00:02:34.110
So principal amortization just
means, well, if I just keep

00:02:34.110 --> 00:02:37.370
paying this mortgage, after
how long is the entire

00:02:37.370 --> 00:02:39.510
principal amount -- not just
interest, how long is the

00:02:39.510 --> 00:02:41.430
entire principal amount going
to be paid off in?

00:02:41.430 --> 00:02:46.410
So essentially, a 30-year fixed
rate loan has a 30 year

00:02:46.410 --> 00:02:47.780
principal amortization.

00:02:47.780 --> 00:02:50.200
If you have a 10-year loan,
you'd put 10 here.

00:02:50.200 --> 00:02:52.350
This is the property tax rate.

00:02:52.350 --> 00:02:55.030
This is what I assume because
I live in California, and in

00:02:55.030 --> 00:02:58.500
most areas of California,
that's the property tax.

00:02:58.500 --> 00:03:00.140
This is what I assume about
annual maintenance.

00:03:00.140 --> 00:03:01.390
That's just an assumption.

00:03:01.390 --> 00:03:04.010
Some houses might be less,
might be more.

00:03:04.010 --> 00:03:05.660
That's up to you to decide.

00:03:05.660 --> 00:03:07.280
This is housing association
dues.

00:03:07.280 --> 00:03:09.900
Maybe if you live in a community
that has a shared

00:03:09.900 --> 00:03:12.500
golf course, or a shared
pool or something.

00:03:12.500 --> 00:03:14.500
Put it at 0 if you don't.

00:03:14.500 --> 00:03:18.360
This is annual insurance, for
things like hazard insurance,

00:03:18.360 --> 00:03:20.600
and flood insurance, earthquake
insurance, or

00:03:20.600 --> 00:03:22.530
whatever insurance you
need where you live.

00:03:22.530 --> 00:03:24.770
And in this bright yellow, I
say, what is the assumed

00:03:24.770 --> 00:03:27.620
annual appreciation of
the house itself?

00:03:27.620 --> 00:03:28.650
And this is a huge assumption.

00:03:28.650 --> 00:03:33.290
And that's why I put it into
this bold yellow color.

00:03:33.290 --> 00:03:35.540
Because we'll see later in
this video that to some

00:03:35.540 --> 00:03:41.245
degree, that assumption is one
of the biggest drivers of

00:03:41.245 --> 00:03:41.600
assumption.

00:03:41.600 --> 00:03:43.880
Or you could say the model is
very, very, very sensitive to

00:03:43.880 --> 00:03:45.390
that assumption.

00:03:45.390 --> 00:03:49.650
Here, this is your assumed
marginal income tax rate.

00:03:49.650 --> 00:03:51.190
And why does that matter?

00:03:51.190 --> 00:03:55.140
Well because you can deduct the
interest that you spend on

00:03:55.140 --> 00:03:56.520
your mortgage.

00:03:56.520 --> 00:03:59.380
And also you can deduct,
actually, the property tax.

00:03:59.380 --> 00:04:04.690
So if you can deduct $100 in
interest and property tax, if

00:04:04.690 --> 00:04:08.850
your marginal tax rate is 30% --
so that means at what rate

00:04:08.850 --> 00:04:10.720
are you being taxed on every
incremental dollar.

00:04:10.720 --> 00:04:14.540
If it's 30%, that means a $100
deduction will save you $30.

00:04:14.540 --> 00:04:17.740
If your marginal tax rate is
20%, a $100 deduction will

00:04:17.740 --> 00:04:18.579
save you $20.

00:04:18.579 --> 00:04:21.060
So that's where that
comes into play.

00:04:21.060 --> 00:04:23.360
The 2%, that's general
inflation.

00:04:23.360 --> 00:04:26.890
And what this assumption drives
is, well, there's going

00:04:26.890 --> 00:04:30.540
to be some inflation on things
like housing association dues,

00:04:30.540 --> 00:04:32.610
annual maintenance, insurance.

00:04:32.610 --> 00:04:34.890
And so this, what you assume
about, well, what is just the

00:04:34.890 --> 00:04:37.460
general rate of inflation, in
our model that's actually

00:04:37.460 --> 00:04:41.360
going to drive how these grow
over the life of your loan.

00:04:41.360 --> 00:04:44.580
And then once you type in all
of these things, the monthly

00:04:44.580 --> 00:04:46.650
mortgage payment
is calculated.

00:04:46.650 --> 00:04:51.140
I assumed that the interest
compounds once a month.

00:04:51.140 --> 00:04:54.570
You can, if you know your
geometric series, you can go

00:04:54.570 --> 00:04:56.730
in there and you can tweak it
around so it compounds more

00:04:56.730 --> 00:04:57.920
frequently or less frequently.

00:04:57.920 --> 00:05:00.070
But my understanding
is that most

00:05:00.070 --> 00:05:02.470
mortgages compound monthly.

00:05:02.470 --> 00:05:05.560
And then this right here, so
this is everything that's

00:05:05.560 --> 00:05:09.480
driving the buying
a home decision.

00:05:09.480 --> 00:05:13.260
Now these assumptions are so
that we can make a comparison

00:05:13.260 --> 00:05:17.010
to, well, what if instead of
using that down payment to buy

00:05:17.010 --> 00:05:21.210
a house, what if we actually
just save that down payment,

00:05:21.210 --> 00:05:24.050
put it in the bank, and
rent a house instead.

00:05:24.050 --> 00:05:28.460
So this is cost of renting
a similar home.

00:05:28.460 --> 00:05:31.580
This is the annual rental
price inflation.

00:05:31.580 --> 00:05:34.060
And I would argue, to some
degree, that rental price

00:05:34.060 --> 00:05:37.140
inflation over the long term
should not be that different

00:05:37.140 --> 00:05:38.640
than housing price inflation.

00:05:38.640 --> 00:05:41.350
Because to some degree,
rental is kind of the

00:05:41.350 --> 00:05:43.120
earnings on a home.

00:05:43.120 --> 00:05:47.090
And if earnings increase and
the overall asset doesn't

00:05:47.090 --> 00:05:48.470
increase, your return
increases.

00:05:48.470 --> 00:05:49.440
Or the other way around.

00:05:49.440 --> 00:05:50.480
Your return decreases.

00:05:50.480 --> 00:05:52.800
But anyway, don't want to
get too complicated.

00:05:52.800 --> 00:05:56.710
And then this is the 6%, or
I just assume it's 6%.

00:05:56.710 --> 00:05:57.320
You can change it.

00:05:57.320 --> 00:06:01.690
This is what you assume that
you can get on your cash.

00:06:01.690 --> 00:06:07.870
So if I don't put the $150,000
down deposit on the home, and

00:06:07.870 --> 00:06:10.690
I put it in, I don't know, maybe
I'm a good investor.

00:06:10.690 --> 00:06:11.700
I could put in the
stock market.

00:06:11.700 --> 00:06:14.160
Maybe I can get 20% a year.

00:06:14.160 --> 00:06:17.920
Or maybe I'm really risk
averse, and I put it in

00:06:17.920 --> 00:06:20.720
government bonds, and
I get 4% a year.

00:06:20.720 --> 00:06:22.990
So this is the assumption
that you get in on that.

00:06:22.990 --> 00:06:27.260
And it actually should be an
after-tax return on that cash.

00:06:27.260 --> 00:06:33.350
So if my tax rate is 30% and I
think I can get 10% percent on

00:06:33.350 --> 00:06:37.460
the stock market, I should
actually put a 7% here.

00:06:37.460 --> 00:06:39.840
So we want to make sure
that we're completely

00:06:39.840 --> 00:06:41.950
accurate for taxes.

00:06:41.950 --> 00:06:45.800
So now let me explain the rest
of the model to you.

00:06:45.800 --> 00:06:51.380
I want to make sure that I can
fit it all within this window.

00:06:51.380 --> 00:06:54.290
Let me just squeeze
this a little bit.

00:06:58.690 --> 00:07:02.830
Excel on YouTube is a
new thing for me.

00:07:02.830 --> 00:07:06.150
That's not what I
wanted to do.

00:07:06.150 --> 00:07:10.060
So let me unfreeze the window.

00:07:10.060 --> 00:07:11.100
OK.

00:07:11.100 --> 00:07:12.430
So now I can show you
the rest of model.

00:07:12.430 --> 00:07:14.610
So all those assumptions
that we did,

00:07:14.610 --> 00:07:16.290
that drives this model.

00:07:16.290 --> 00:07:17.940
Let me freeze the window
right here.

00:07:21.580 --> 00:07:21.790
OK.

00:07:21.790 --> 00:07:25.220
That should make things
a little bit easier.

00:07:25.220 --> 00:07:28.540
So this is the buying scenario,
up to line 40.

00:07:28.540 --> 00:07:31.950
This says, OK, at period zero,
what is the home value?

00:07:31.950 --> 00:07:33.210
And don't type in
anything here.

00:07:33.210 --> 00:07:35.170
It's all automatically
calculated.

00:07:35.170 --> 00:07:37.340
So at period zero, what
is your home value?

00:07:37.340 --> 00:07:40.580
And then it uses essentially
the appreciation numbers.

00:07:40.580 --> 00:07:42.050
And each period is essentially
a month.

00:07:42.050 --> 00:07:43.680
I actually wrote
that down here.

00:07:43.680 --> 00:07:46.050
And then it figures out,
what is the market

00:07:46.050 --> 00:07:46.500
value of your home?

00:07:46.500 --> 00:07:49.420
And it's completely driven by
that appreciation number.

00:07:49.420 --> 00:07:53.420
This right here is the debt, or
essentially the principal

00:07:53.420 --> 00:07:56.210
payment on your mortgage,
or how much do

00:07:56.210 --> 00:07:57.410
you owe to the bank.

00:07:57.410 --> 00:08:01.710
And as you see, as months go by,
when you pay the mortgage

00:08:01.710 --> 00:08:04.510
note -- and I show that right
here, this mortgage payment.

00:08:04.510 --> 00:08:07.870
Some amount of that, which is
line 33, the principal paid.

00:08:07.870 --> 00:08:11.480
Some of that goes to decrease
the amount you owe.

00:08:11.480 --> 00:08:14.230
And then a lot of it, especially
initially, goes to

00:08:14.230 --> 00:08:16.700
be actually the interest
on the amount you owe.

00:08:16.700 --> 00:08:20.590
And then obviously, if you
watch the video on

00:08:20.590 --> 00:08:22.480
introduction to balance sheets,
your equity in the

00:08:22.480 --> 00:08:27.170
home is the value of the home
minus the debt, or minus what

00:08:27.170 --> 00:08:27.880
you owe the bank.

00:08:27.880 --> 00:08:29.660
So this actually calculates
your equity.

00:08:29.660 --> 00:08:31.450
Or essentially, one way to view
it, is actually to say,

00:08:31.450 --> 00:08:33.900
well, what am I worth?

00:08:33.900 --> 00:08:37.539
Or what is this investment worth
to me at that point?

00:08:37.539 --> 00:08:39.730
So these are kind of the
important numbers in the home

00:08:39.730 --> 00:08:40.190
buying scenario.

00:08:40.190 --> 00:08:42.780
It is driven by-- This interest
on debt, it's

00:08:42.780 --> 00:08:46.910
calculated by what interest
rate you assume, times the

00:08:46.910 --> 00:08:48.930
debt you owe and the
period before.

00:08:48.930 --> 00:08:51.440
The mortgage payment, we
calculated that before, using

00:08:51.440 --> 00:08:54.820
our mathematical knowledge
of geometric series.

00:08:54.820 --> 00:08:57.440
The paid principal, that's
going to be the mortgage

00:08:57.440 --> 00:09:01.070
payment minus your interest.
Insurance payment, it's on a

00:09:01.070 --> 00:09:02.300
monthly basis, right?

00:09:02.300 --> 00:09:05.270
So we essentially took whatever
our annual insurance

00:09:05.270 --> 00:09:06.940
payment was and we
divide it by 12.

00:09:06.940 --> 00:09:09.720
But then we grow it by
the rate of inflation

00:09:09.720 --> 00:09:10.760
on a monthly basis.

00:09:10.760 --> 00:09:12.580
So we took the inflation rate,
divided by 12, and we

00:09:12.580 --> 00:09:15.340
multiplied by each
of these months.

00:09:15.340 --> 00:09:18.440
The housing association dues,
once again, this is on a

00:09:18.440 --> 00:09:19.050
monthly basis.

00:09:19.050 --> 00:09:21.230
So we just took your assumption,
divided by 12.

00:09:21.230 --> 00:09:22.510
Maintenance, same thing.

00:09:22.510 --> 00:09:24.660
Property tax, same thing.

00:09:24.660 --> 00:09:27.910
Although I assumed that your
house gets reassessed.

00:09:27.910 --> 00:09:32.610
So you're in a state where every
year, or every several

00:09:32.610 --> 00:09:34.310
years, the assessor comes, says,
oh, your house is worth

00:09:34.310 --> 00:09:35.510
more now, so I'm going
to raise your taxes.

00:09:35.510 --> 00:09:37.460
That's not the case in a lot
of parts of California.

00:09:37.460 --> 00:09:40.970
But it's the case in many parts
of the U.S. So actually

00:09:40.970 --> 00:09:43.950
to some degree, this, the dollar
value, the property tax

00:09:43.950 --> 00:09:47.410
is driven by this home value
assumption up here.

00:09:47.410 --> 00:09:50.230
This income tax saving from
interest deduction, this is

00:09:50.230 --> 00:09:52.710
assuming that at that marginal
tax rate, you can deduct the

00:09:52.710 --> 00:09:55.700
property tax and the interest
on the debt.

00:09:55.700 --> 00:09:59.810
And then this is the total cash
outflow after adding back

00:09:59.810 --> 00:10:00.860
the income tax savings.

00:10:00.860 --> 00:10:03.880
So this is essentially how much
cash goes out the door,

00:10:03.880 --> 00:10:06.740
even after the tax savings,
every month,

00:10:06.740 --> 00:10:08.240
in the buying scenario.

00:10:08.240 --> 00:10:09.830
That's what that is.

00:10:09.830 --> 00:10:11.270
So hopefully that makes
a little bit of sense.

00:10:11.270 --> 00:10:13.780
So what we want to do is, we
want to figure out, OK, you

00:10:13.780 --> 00:10:14.520
could do that.

00:10:14.520 --> 00:10:18.830
You could buy the house,
put $150,000 down.

00:10:18.830 --> 00:10:22.400
And every month put this much
out, and as you see that

00:10:22.400 --> 00:10:23.270
number grows.

00:10:23.270 --> 00:10:25.755
The mortgage is the same, but
a lot of these expenses grow

00:10:25.755 --> 00:10:26.890
with inflation.

00:10:26.890 --> 00:10:28.980
But I want to compare that to
what happens if I take that

00:10:28.980 --> 00:10:31.730
exact amount of cash, after
adjusting for how much money I

00:10:31.730 --> 00:10:33.170
get back from taxes.

00:10:33.170 --> 00:10:37.490
And if I said, well, I'm going
to use that cash to pay my

00:10:37.490 --> 00:10:40.440
rent and any other expenses
associated with renting--

00:10:40.440 --> 00:10:43.810
which really aren't much-- to
pay my rent, and then put the

00:10:43.810 --> 00:10:45.470
rest in the bank.

00:10:45.470 --> 00:10:48.430
So what we're saying is, well,
that assumption was, that you

00:10:48.430 --> 00:10:49.950
can rent a similar
house for $2,500.

00:10:49.950 --> 00:10:51.690
It may be right, it
may be wrong.

00:10:51.690 --> 00:10:53.860
It's up for you to play with.

00:10:53.860 --> 00:10:56.440
And of course it grows with
inflation slowly.

00:10:56.440 --> 00:10:58.140
Obviously your rent doesn't
increase every month, but I

00:10:58.140 --> 00:10:59.830
assume it does fairly
continuously.

00:10:59.830 --> 00:11:01.260
It's a reasonable assumption
I think.

00:11:01.260 --> 00:11:02.110
Although you can change it.

00:11:02.110 --> 00:11:05.880
You can make it only
step up every year.

00:11:05.880 --> 00:11:08.700
And then this line down
here tells us the

00:11:08.700 --> 00:11:12.100
savings while renting.

00:11:12.100 --> 00:11:14.810
And I'm not saying the savings
from, you know, something's on

00:11:14.810 --> 00:11:15.920
sale so I save money.

00:11:15.920 --> 00:11:18.660
But your savings in terms of how
much you have in the bank.

00:11:18.660 --> 00:11:21.370
So if you rented instead of
putting that $150,000 as a

00:11:21.370 --> 00:11:23.170
down payment, you could have
put it in the bank.

00:11:23.170 --> 00:11:25.590
So that would be your savings
account at period zero.

00:11:25.590 --> 00:11:28.660
And then your savings account
at period one would be this

00:11:28.660 --> 00:11:31.820
amount of money and whatever
return you got it, plus the

00:11:31.820 --> 00:11:34.730
difference between your
cash out from buying a

00:11:34.730 --> 00:11:36.860
home and your rent.

00:11:36.860 --> 00:11:38.670
So this is your savings.

00:11:38.670 --> 00:11:43.650
So what I do in this model --
and I could show you, I could

00:11:43.650 --> 00:11:47.670
scroll through multiple
periods.

00:11:47.670 --> 00:11:51.710
Yeah, this model actually goes
as far as Excel would let me.

00:11:51.710 --> 00:11:55.160
But the average house -- anyone
who's traded mortgage

00:11:55.160 --> 00:11:59.540
bonds will tell you-- the
average mortgage loan has a 10

00:11:59.540 --> 00:12:01.260
year expected life.

00:12:01.260 --> 00:12:03.135
Because that's when, on average,
people tend to move

00:12:03.135 --> 00:12:05.000
or refinance.

00:12:05.000 --> 00:12:08.957
So what I do is I figure out,
well, given your assumptions

00:12:08.957 --> 00:12:11.720
-- you can make your own
assumptions -- given your

00:12:11.720 --> 00:12:16.880
assumptions, what is
your home value?

00:12:16.880 --> 00:12:18.890
So let me make sure
I can get to that.

00:12:21.680 --> 00:12:24.360
So given your assumptions, what
this calculates is, well,

00:12:24.360 --> 00:12:27.250
it tells you what the home value
is after 10 years, your

00:12:27.250 --> 00:12:30.150
debt after 10 years, your home
equity after 10 years.

00:12:30.150 --> 00:12:31.980
And it assumes you were
to sell your house.

00:12:31.980 --> 00:12:33.630
Because that's what the
average American

00:12:33.630 --> 00:12:34.710
does after 10 years.

00:12:34.710 --> 00:12:36.370
And so what is the
transaction cost?

00:12:36.370 --> 00:12:37.770
You pay 6% to a broker.

00:12:37.770 --> 00:12:41.290
Hopefully that won't be the
case in 10 years and the

00:12:41.290 --> 00:12:43.050
internet will dis-intermediate
real estate

00:12:43.050 --> 00:12:44.190
brokers, but who knows.

00:12:44.190 --> 00:12:46.780
I apologize to if
you are broker.

00:12:46.780 --> 00:12:52.410
And then this line, line 54,
that tells you what the net

00:12:52.410 --> 00:12:54.470
cash is if you sell your
house at a market

00:12:54.470 --> 00:12:56.800
price, you pay the broker.

00:12:56.800 --> 00:12:59.380
This number right here is much
simpler to some degree.

00:12:59.380 --> 00:13:01.410
It just tells you, well
let's say you decided

00:13:01.410 --> 00:13:02.270
not to buy the house.

00:13:02.270 --> 00:13:05.160
Given all your assumptions, how
much would you have saved

00:13:05.160 --> 00:13:07.360
in the bank at that time?

00:13:07.360 --> 00:13:11.110
And so this number right here,
this number is the difference

00:13:11.110 --> 00:13:14.030
between those two numbers
in 10 years,

00:13:14.030 --> 00:13:16.276
discounted back to today.

00:13:19.670 --> 00:13:22.390
Actually I meant to
present value it.

00:13:22.390 --> 00:13:25.860
But did I present value
these numbers?

00:13:25.860 --> 00:13:26.790
Oh no, I didn't.

00:13:26.790 --> 00:13:28.120
So actually this was meant
to be the present value.

00:13:28.120 --> 00:13:30.040
I'm going to correct that
before you actually

00:13:30.040 --> 00:13:30.560
play with the model.

00:13:30.560 --> 00:13:32.990
Right now I just took the 10
year value, so this is the

00:13:32.990 --> 00:13:33.980
value in year 10.

00:13:33.980 --> 00:13:35.450
This is the difference
between the two.

00:13:35.450 --> 00:13:38.230
The present value would be if
you discount this by some

00:13:38.230 --> 00:13:38.900
discount rate.

00:13:38.900 --> 00:13:41.080
Whatever you think, probably
the inflation rate.

00:13:41.080 --> 00:13:43.240
And it would tell you in today's
money, what is the

00:13:43.240 --> 00:13:46.480
benefit or the advantage of
buying versus renting?

00:13:46.480 --> 00:13:48.600
Anyway I've spent 14 minutes
of your time.

00:13:48.600 --> 00:13:52.610
I encourage you to download this
model, play with it, and

00:13:52.610 --> 00:13:53.640
then work out the assumptions.

00:13:53.640 --> 00:13:55.690
Because I think that's
the important thing.

00:13:55.690 --> 00:13:57.150
Some people, they'll
make some set of

00:13:57.150 --> 00:13:58.810
assumptions and say, ah-ha!

00:13:58.810 --> 00:13:59.670
I should rent.

00:13:59.670 --> 00:14:00.730
Or they say, ah-ha!

00:14:00.730 --> 00:14:01.360
I should buy.

00:14:01.360 --> 00:14:04.220
But they don't realize that they
made some assumptions.

00:14:04.220 --> 00:14:07.630
That although it looks really
reasonable, let's say I make

00:14:07.630 --> 00:14:10.870
this 3% annual appreciation
assumption.

00:14:10.870 --> 00:14:12.410
That doesn't seem crazy.

00:14:12.410 --> 00:14:15.060
But it's amazing how much it'll
change the model if you

00:14:15.060 --> 00:14:19.750
make that 3% into a 1%, or if
you make it into even a

00:14:19.750 --> 00:14:21.280
negative 1% or negative 2%.

00:14:21.280 --> 00:14:22.180
It's completely possible.

00:14:22.180 --> 00:14:25.660
It's happened before in the past
that you have flat real

00:14:25.660 --> 00:14:28.360
estate prices for a significant
period of time,

00:14:28.360 --> 00:14:29.880
even 10 years.

00:14:29.880 --> 00:14:32.410
And actually most of the studies
show that real estate,

00:14:32.410 --> 00:14:35.350
over the last 100 years, has
actually roughly grown, in

00:14:35.350 --> 00:14:38.000
real terms, maybe 1% or 2%.

00:14:38.000 --> 00:14:41.910
So actually 1% or 2% percent
here isn't that conservative.

00:14:41.910 --> 00:14:43.560
And actually especially after
a big real estate

00:14:43.560 --> 00:14:46.470
boom, may be prudent.

00:14:46.470 --> 00:14:47.950
So play with these
assumptions.

00:14:47.950 --> 00:14:49.870
And I think it'll give you
an intuition of what

00:14:49.870 --> 00:14:50.790
are the real drivers.

00:14:50.790 --> 00:14:53.250
Another big thing -- sometimes
you don't rent a similar home.

00:14:53.250 --> 00:14:54.210
You'd rent a smaller home.

00:14:54.210 --> 00:14:55.590
So that would be a different
type of savings.

00:14:55.590 --> 00:14:56.770
And there are trade-offs
there.

00:14:56.770 --> 00:15:00.380
But anyway, hopefully you'll
find this model useful.

00:15:00.380 --> 00:15:03.240
I think it should be.

00:15:03.240 --> 00:15:05.020
People, this is the biggest
investment of their life.

00:15:05.020 --> 00:15:08.290
They should do serious analysis
when they think about

00:15:08.290 --> 00:15:09.270
how they want approach it.

00:15:09.270 --> 00:15:10.810
And I'd like to think
that this is

00:15:10.810 --> 00:15:12.180
fairly serious analysis.

00:15:12.180 --> 00:15:14.540
This is about as serious
as you can get.

00:15:14.540 --> 00:15:15.450
So enjoy!

00:15:15.450 --> 00:15:17.250
See you in the next video.


Proof: Field from infinite plate (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:05.240
So where I left off, we had
this infinite plate.

00:00:05.240 --> 00:00:07.900
It's just an infinite plane, and
it's a charged plate with

00:00:07.900 --> 00:00:09.410
a charge density sigma.

00:00:09.410 --> 00:00:12.830
And what we did is we said, OK,
well, we're taking this

00:00:12.830 --> 00:00:16.140
point up here that's h units
above the surface of our

00:00:16.140 --> 00:00:20.300
charge plate, and we wanted to
figure out the electric field

00:00:20.300 --> 00:00:25.150
at that point, generated by a
ring of radius r essentially

00:00:25.150 --> 00:00:31.400
centered at the base of where
that point is above.

00:00:31.400 --> 00:00:33.830
We want to figure out what is
the electric field generated

00:00:33.830 --> 00:00:36.130
by this ring at that point?

00:00:36.130 --> 00:00:42.450
And we figured out that the
electric field was this, and

00:00:42.450 --> 00:00:44.900
then because we made a symmetry
argument in the last

00:00:44.900 --> 00:00:47.110
video, we only care about
the y-component.

00:00:47.110 --> 00:00:49.700
Because we figured out that at
the electric field generated

00:00:49.700 --> 00:00:52.510
from any point, the x-components
cancel out,

00:00:52.510 --> 00:00:55.295
because if we have a point
here, it'll have some

00:00:55.295 --> 00:00:55.560
x-component.

00:00:55.560 --> 00:00:58.710
The field's x-component might
be in that direction to the

00:00:58.710 --> 00:01:00.650
right, but then you have another
point out here, and

00:01:00.650 --> 00:01:03.130
its x-component will
just cancel it out.

00:01:03.130 --> 00:01:05.760
So we only care about
the y-component.

00:01:05.760 --> 00:01:10.280
So at the end, we meticulously
calculated what the

00:01:10.280 --> 00:01:15.110
y-component of the electric
field generated by the ring

00:01:15.110 --> 00:01:18.570
is, at h units above
the surface.

00:01:18.570 --> 00:01:22.570
So with that out of the way,
let's see if we can sum up a

00:01:22.570 --> 00:01:26.480
bunch of rings going from radius
infinity to radius zero

00:01:26.480 --> 00:01:28.950
and figure out the total
y-component.

00:01:28.950 --> 00:01:30.750
Or essentially the total
electric field, because we

00:01:30.750 --> 00:01:33.490
realize that all the x's cancel
out anyway, the total

00:01:33.490 --> 00:01:37.120
electric field at that point,
h units above the

00:01:37.120 --> 00:01:39.930
surface of the plane.

00:01:39.930 --> 00:01:46.110
So let me erase a lot of this
just so I can free it up for

00:01:46.110 --> 00:01:49.405
some hard-core math.

00:01:49.405 --> 00:01:54.130
And this is pretty much all
calculus at this point.

00:01:54.130 --> 00:01:55.640
So let me erase all of this.

00:01:55.640 --> 00:01:58.090
Watch the previous video if you
forgot how it was derived.

00:02:01.370 --> 00:02:03.380
Let me even erase that because
I think I will

00:02:03.380 --> 00:02:04.630
need a lot of space.

00:02:07.400 --> 00:02:09.220
There you go.

00:02:09.220 --> 00:02:12.770
OK, so let me redraw a little
bit just so we never forget

00:02:12.770 --> 00:02:17.390
what we're doing here because
that happens.

00:02:17.390 --> 00:02:20.390
So that's my plane that goes
off in every direction.

00:02:25.110 --> 00:02:28.050
I have my point above the plane
where we're trying to

00:02:28.050 --> 00:02:29.130
figure out the electric field.

00:02:29.130 --> 00:02:30.740
And we've come to the conclusion
that the field is

00:02:30.740 --> 00:02:32.640
going to point upward, so
we only care about the

00:02:32.640 --> 00:02:33.860
y-component.

00:02:33.860 --> 00:02:39.590
It's h units above the surface,
and we're figuring

00:02:39.590 --> 00:02:43.200
out the electric field generated
by a ring around

00:02:43.200 --> 00:02:46.630
this point of radius r.

00:02:46.630 --> 00:02:49.010
And what's the y-component
of that electric field?

00:02:49.010 --> 00:02:50.760
We figured out it was this.

00:02:50.760 --> 00:02:53.230
So now what we're going to
do is take the integral.

00:02:53.230 --> 00:03:00.870
So the total electric field from
the plate is going to be

00:03:00.870 --> 00:03:06.440
the integral from-- that's a
really ugly-looking integral--

00:03:06.440 --> 00:03:09.620
a radius of zero to a
radius of infinity.

00:03:09.620 --> 00:03:11.970
So we're going to take a sum of
all of the rings, starting

00:03:11.970 --> 00:03:14.960
with a radius of zero all the
way to the ring that has a

00:03:14.960 --> 00:03:16.620
radius of infinity, because
it's an infinite plane so

00:03:16.620 --> 00:03:20.180
we're figuring out the impact
of the entire plane.

00:03:20.180 --> 00:03:24.060
So we're going to take the sum
of every ring, so the field

00:03:24.060 --> 00:03:26.240
generated by every ring, and
this is the field generated by

00:03:26.240 --> 00:03:28.275
each of the rings.

00:03:28.275 --> 00:03:29.710
Let me do it in a
different color.

00:03:29.710 --> 00:03:32.340
This light blue is getting
a little monotonous.

00:03:32.340 --> 00:03:47.050
Kh 2pi sigma r dr over
h squared plus r

00:03:47.050 --> 00:03:50.030
squared to the 3/2.

00:03:50.030 --> 00:03:51.280
Now, let's simplify
this a little bit.

00:03:51.280 --> 00:03:54.040
Let's take some constants out of
it just so this looks like

00:03:54.040 --> 00:03:56.410
a slightly simpler equation.

00:03:56.410 --> 00:04:04.795
So this equals the integral from
zero to-- So let's take

00:04:04.795 --> 00:04:06.670
the K-- I'm going to
leave the 2 there.

00:04:06.670 --> 00:04:08.370
You'll see why in a second, but
I'm going to take all the

00:04:08.370 --> 00:04:10.770
other constants out that we're
not integrating across.

00:04:10.770 --> 00:04:23.690
So it's equal to Kh pi sigma
times the integral from zero

00:04:23.690 --> 00:04:27.270
to infinity of what is this?

00:04:27.270 --> 00:04:28.930
So what did I leave in there?

00:04:28.930 --> 00:04:35.480
I left a 2r, so we could
rewrite this as-- well,

00:04:35.480 --> 00:04:37.560
actually, I'm running
out of space.

00:04:37.560 --> 00:04:50.470
2r dr over h squared plus r
squared to the 3/2, or we

00:04:50.470 --> 00:04:53.380
could think of it as the
negative 3/2, right?

00:04:53.380 --> 00:04:55.380
So what is the antiderivative
of here?

00:04:55.380 --> 00:04:58.800
Well, this is essentially the
reverse chain rule, right?

00:04:58.800 --> 00:05:00.850
I could make a substitution
here, if you're more

00:05:00.850 --> 00:05:03.390
comfortable using the
substitution rule, but you

00:05:03.390 --> 00:05:04.950
might be able to eyeball
this at this point.

00:05:04.950 --> 00:05:12.390
We could make the substitution
that u is equal to-- if we

00:05:12.390 --> 00:05:14.530
just want to figure out the
antiderivative of this-- if u

00:05:14.530 --> 00:05:18.300
is equal to h squared plus
r squared-- h is just a

00:05:18.300 --> 00:05:25.100
constant, right-- then du is
just equal to-- I mean, the du

00:05:25.100 --> 00:05:33.610
dr-- this is a constant, so it
equals 2r, or we could say du

00:05:33.610 --> 00:05:36.670
is equal to 2r dr.

00:05:36.670 --> 00:05:45.490
And so if we're trying to take
the antiderivative of 2r dr

00:05:45.490 --> 00:05:53.370
over h squared plus r squared to
the 3/2, this is the exact

00:05:53.370 --> 00:05:56.320
same thing as taking the
antiderivative with this

00:05:56.320 --> 00:05:57.310
substitution.

00:05:57.310 --> 00:05:59.680
2r dr, we just showed right
here, that's the same

00:05:59.680 --> 00:06:01.650
thing as du, right?

00:06:01.650 --> 00:06:06.100
So that's du over-- and then
this is just u, right?

00:06:06.100 --> 00:06:07.370
H squared plus r squared is u.

00:06:07.370 --> 00:06:08.590
We do that by definition.

00:06:08.590 --> 00:06:13.350
So u to the 3/2, which is equal
to the antiderivative

00:06:13.350 --> 00:06:20.260
of-- we could write this as
u to the minus 3/2 du.

00:06:20.260 --> 00:06:20.960
And now that's easy.

00:06:20.960 --> 00:06:25.100
This is just kind of reverse
the exponent rule.

00:06:25.100 --> 00:06:32.710
So that equals minus 2u to
the minus 1/2, and we

00:06:32.710 --> 00:06:33.530
can confirm, right?

00:06:33.530 --> 00:06:35.750
If we take the derivative of
this, minus 1/2 times minus 2

00:06:35.750 --> 00:06:40.320
is 1, and then subtract 1 from
here, we get minus 3/2.

00:06:40.320 --> 00:06:42.440
And then we could add plus c,
but since we're eventually

00:06:42.440 --> 00:06:43.590
going to do a definite
integral, the

00:06:43.590 --> 00:06:45.290
c's all cancel out.

00:06:45.290 --> 00:06:48.600
Or we could say that this is
equal to-- since we made that

00:06:48.600 --> 00:06:57.950
substitution-- minus 2 over--
minus 1/2, that's the same

00:06:57.950 --> 00:07:03.170
thing as over the square
root of h squared

00:07:03.170 --> 00:07:05.570
plus r squared, right?

00:07:05.570 --> 00:07:08.290
So all of the stuff I did in
magenta was just to figure out

00:07:08.290 --> 00:07:11.330
the antiderivative of this, and
we figured it out to be

00:07:11.330 --> 00:07:18.270
this: minus 2 over the
square root of h

00:07:18.270 --> 00:07:19.220
squared plus r squared.

00:07:19.220 --> 00:07:22.760
So with that out of the way,
let's continue evaluating our

00:07:22.760 --> 00:07:25.020
definite integral.

00:07:25.020 --> 00:07:29.170
So this expression simplifies
to-- this is a marathon

00:07:29.170 --> 00:07:33.920
problem, but satisfying-- K--
let's get all the constants--

00:07:33.920 --> 00:07:43.320
Kh pi sigma-- we can even take
this minus 2 out-- times minus

00:07:43.320 --> 00:07:46.970
2, and all of that, and we're
going to evaluate the definite

00:07:46.970 --> 00:07:53.910
integral at the two boundaries--
1 over the square

00:07:53.910 --> 00:08:00.700
root of h squared plus r squared
evaluated at infinity

00:08:00.700 --> 00:08:05.040
minus it evaluated
at 0, right?

00:08:05.040 --> 00:08:09.150
Well, what does this
expression equal?

00:08:09.150 --> 00:08:11.840
What is 1 over the square
root of h squared

00:08:11.840 --> 00:08:13.280
plus infinity, right?

00:08:13.280 --> 00:08:15.210
What happens when we evaluate
r at infinity?

00:08:15.210 --> 00:08:18.420
Well, the square root of
infinity is still infinity,

00:08:18.420 --> 00:08:21.955
and 1 over infinity is 0, so
this expression right here

00:08:21.955 --> 00:08:23.140
just becomes 0.

00:08:23.140 --> 00:08:27.500
When you evaluate it at
infinity, this becomes 0 minus

00:08:27.500 --> 00:08:29.750
this expression evaluated
at 0.

00:08:29.750 --> 00:08:31.670
So what happens when
it's at 0?

00:08:31.670 --> 00:08:35.710
When r squared is 0, we get 1
over the square root of h

00:08:35.710 --> 00:08:37.960
squared, right?

00:08:37.960 --> 00:08:39.080
So let's write it all out.

00:08:39.080 --> 00:08:47.730
This becomes minus 2Kh pi sigma
times 0 minus 1 over the

00:08:47.730 --> 00:08:49.140
square root of h squared.

00:08:49.140 --> 00:09:01.870
Well this equals minus 2Kh pi
sigma times-- well, 1 over the

00:09:01.870 --> 00:09:05.080
square root of h squared, that's
just 1 over h, right?

00:09:05.080 --> 00:09:09.130
And there's a minus times
minus 1 over h.

00:09:09.130 --> 00:09:12.820
Well, this minus and that
minus cancel out.

00:09:12.820 --> 00:09:17.730
And then this h and this 1
over h should cancel out.

00:09:17.730 --> 00:09:21.170
And all we're left with, after
doing all of that work, and

00:09:21.170 --> 00:09:23.590
I'll do it in a bright color
because we've done a lot of

00:09:23.590 --> 00:09:28.760
work to get here,
is 2K pi sigma.

00:09:31.380 --> 00:09:33.150
So let's see it at
a lot of levels.

00:09:33.150 --> 00:09:35.270
First of all, what did
we even do here?

00:09:35.270 --> 00:09:37.010
We might have gotten
lost in the math.

00:09:37.010 --> 00:09:41.640
This is the net electric, the
total electric field, at a

00:09:41.640 --> 00:09:48.970
point at height h above this
infinite plate that has a

00:09:48.970 --> 00:09:52.390
uniform charge, and the charge
density is sigma.

00:09:52.390 --> 00:09:55.250
But notice, this is the electric
field at that point,

00:09:55.250 --> 00:09:56.700
but there's no h in here.

00:09:56.700 --> 00:10:00.430
So it essentially is telling
us that the strength of the

00:10:00.430 --> 00:10:05.320
field is in no way dependent on
how high above the field we

00:10:05.320 --> 00:10:08.160
are, which tells us this is
going to be a constant field.

00:10:08.160 --> 00:10:10.020
We can be anywhere above
the plate and the

00:10:10.020 --> 00:10:11.540
charge will be the same.

00:10:11.540 --> 00:10:13.830
The only thing-- oh, sorry,
not the charge.

00:10:13.830 --> 00:10:18.600
The field will be the same, and
if we have a test charge,

00:10:18.600 --> 00:10:20.110
the force would be the same.

00:10:20.110 --> 00:10:23.000
And the only thing that the
strength of the field or the

00:10:23.000 --> 00:10:25.450
strength of the exerted
electrostatic force is

00:10:25.450 --> 00:10:28.090
dependent on, is the charge
density, right?

00:10:28.090 --> 00:10:32.130
This is Coulomb's constant, pi
is pi, 2pi, and I think it's

00:10:32.130 --> 00:10:34.100
kind of cool that it involves
pi, but that's

00:10:34.100 --> 00:10:35.465
something else to ponder.

00:10:35.465 --> 00:10:39.640
But all that matters is
the charge density.

00:10:39.640 --> 00:10:43.210
So hopefully, you found that
reasonably satisfying, and the

00:10:43.210 --> 00:10:46.930
big thing that we learned here
is that if I have an infinite

00:10:46.930 --> 00:10:52.500
uniformly charged plate, the
field-- and I'm some distance

00:10:52.500 --> 00:10:57.860
h above that field-- above that
plate, it doesn't matter

00:10:57.860 --> 00:10:58.750
what that h is.

00:10:58.750 --> 00:11:02.210
I could be here, I could be
here, I could be here.

00:11:02.210 --> 00:11:05.560
At all of those points, the
field has the exact same

00:11:05.560 --> 00:11:08.300
strength, or the net
electrostatic force on a test

00:11:08.300 --> 00:11:11.470
charge at those points has the
exact same strength, and

00:11:11.470 --> 00:11:13.290
that's kind of a neat thing.

00:11:13.290 --> 00:11:15.590
And now if you do believe
everything that occurred in

00:11:15.590 --> 00:11:20.980
the last two videos, you can
now believe that there are

00:11:20.980 --> 00:11:23.400
such things as uniform electric
fields and they occur

00:11:23.400 --> 00:11:25.670
between parallel plates,
especially far away from the

00:11:25.670 --> 00:11:26.770
boundaries.

00:11:26.770 --> 00:11:28.240
See you soon.


Binomial theorem combinatorics connection

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.030
In this video, I'm going to
attempt to give you an

00:00:03.030 --> 00:00:07.000
intuition behind why
multiplying binomials involve

00:00:07.000 --> 00:00:07.670
combinatorics

00:00:07.670 --> 00:00:09.950
Why we actually have the
binomial coefficients

00:00:09.950 --> 00:00:12.240
in there at all.

00:00:12.240 --> 00:00:14.060
And I'm going to do
multiple colors.

00:00:14.060 --> 00:00:18.100
The colors will actually be
non-arbitrary this time.

00:00:18.100 --> 00:00:19.180
Just to give you an intuition.

00:00:19.180 --> 00:00:22.200
So let's multiply a plus
b to the third power.

00:00:22.200 --> 00:00:30.280
Well, a plus b to the third
power, that's a plus b times--

00:00:30.280 --> 00:00:31.540
and I'm going to keep
switching colors.

00:00:31.540 --> 00:00:33.450
You're going to have to bear
with me, but it should

00:00:33.450 --> 00:00:40.260
hopefully be fruitful.-- Times
a plus b, times-- let me pick

00:00:40.260 --> 00:00:44.960
an appropriately different
color, maybe a blue--

00:00:44.960 --> 00:00:50.330
times a plus b.

00:00:50.330 --> 00:00:51.820
And let's do this as a
distributive property.

00:00:51.820 --> 00:01:03.490
This equals a times-- go
back in green-- a plus b.

00:01:03.490 --> 00:01:04.370
With a different green.

00:01:04.370 --> 00:01:08.010
Want to make sure I use the
right green, just because the

00:01:08.010 --> 00:01:09.520
colors matter this time.

00:01:09.520 --> 00:01:16.000
a plus b-- this is tedious,
but it's worth it--

00:01:16.000 --> 00:01:24.560
plus b times a plus b.

00:01:24.560 --> 00:01:34.890
And then all of that
times a plus b again.

00:01:34.890 --> 00:01:38.450
And let's multiply
this inside part.

00:01:38.450 --> 00:01:39.910
I'm just going to keep
it and multiply it out.

00:01:39.910 --> 00:01:49.690
So it's going to be a times
green a-- we know they're all

00:01:49.690 --> 00:01:56.610
the same a-- plus-- I should do
the pluses in a neutral color,

00:01:56.610 --> 00:02:02.050
but it's OK-- plus a times--
you might be finding this

00:02:02.050 --> 00:02:07.300
tedious, but it's going to pay
off in the end-- a times b.

00:02:07.300 --> 00:02:25.400
Then we have plus b times a,
plus yellow b times green b.

00:02:25.400 --> 00:02:27.200
And then all of that--
we're almost there.

00:02:27.200 --> 00:02:28.680
We're almost there.

00:02:28.680 --> 00:02:33.580
All of that times a plus b.

00:02:33.580 --> 00:02:35.820
And it's essentially,
we're going to multiply

00:02:35.820 --> 00:02:37.320
a times everything.

00:02:37.320 --> 00:02:41.220
This blue a times all of
this, and plus this blue

00:02:41.220 --> 00:02:42.080
b times all of this.

00:02:42.080 --> 00:02:44.120
So let's multiply the blue
a times all of this.

00:02:44.120 --> 00:02:46.195
So the first term will
be yellow a, green

00:02:46.195 --> 00:02:48.540
a, and then blue a.

00:02:48.540 --> 00:03:00.450
So it'll be a, a, a.

00:03:00.450 --> 00:03:03.580
Oh it's a different blue, but
I think you get the point.

00:03:03.580 --> 00:03:07.650
Plus this times blue a.

00:03:07.650 --> 00:03:25.260
So yellow a, green b, and then
blue a, plus yellow b, green

00:03:25.260 --> 00:03:31.640
a, blue a-- hopefully I'm
not confusing you-- baa.

00:03:31.640 --> 00:03:32.980
b, a, a.

00:03:32.980 --> 00:03:34.170
We're almost there.

00:03:34.170 --> 00:03:45.510
Plus yellow b times
green b times blue a.

00:03:45.510 --> 00:03:47.880
So we did all the
blue a's, finally.

00:03:47.880 --> 00:03:48.980
There's a blue a.

00:03:52.040 --> 00:03:54.210
Plus--.

00:03:54.210 --> 00:03:57.750
Now we're going to do the
blue b times everything.

00:03:57.750 --> 00:04:03.810
So it's yellow a times green a.

00:04:03.810 --> 00:04:04.030
Right?

00:04:04.030 --> 00:04:10.290
Yellow a, green a, then blue b,
times green a, times blue b--

00:04:10.290 --> 00:04:14.090
almost there, I know this is
tedious-- times blue

00:04:14.090 --> 00:04:16.990
b, plus yellow a.

00:04:16.990 --> 00:04:19.170
Intuition doesn't
come easy, though.

00:04:19.170 --> 00:04:21.710
Yellow a-- so we're on
this term-- yellow

00:04:21.710 --> 00:04:22.930
a, green b, blue b.

00:04:22.930 --> 00:04:27.110
So green b times blue b.

00:04:30.760 --> 00:04:33.810
Now we're at yellow b-- the
good thing about the colors

00:04:33.810 --> 00:04:35.630
is it's easy to keep
track of where we are.

00:04:35.630 --> 00:04:46.980
Plus yellow b times
green a times blue b.

00:04:46.980 --> 00:04:49.090
And then we're at the last one.

00:04:49.090 --> 00:05:01.830
Plus yellow b, green
b, times blue b.

00:05:01.830 --> 00:05:04.070
So this is the expansion
of a plus b to the

00:05:04.070 --> 00:05:04.890
third power, right?

00:05:04.890 --> 00:05:07.190
We haven't simplified
it at all, and I did

00:05:07.190 --> 00:05:07.870
that for a reason.

00:05:07.870 --> 00:05:12.520
Because you see that every term
here-- what's happening here?

00:05:12.520 --> 00:05:17.200
Every term has exactly
one-- it's 3 numbers

00:05:17.200 --> 00:05:18.600
being multiplied, right?

00:05:18.600 --> 00:05:21.210
Every term is 3 numbers
being multiplied.

00:05:21.210 --> 00:05:25.110
And it's one of, you know, the
yellow number comes from the

00:05:25.110 --> 00:05:27.770
first-- from this
yellow a plus b.

00:05:27.770 --> 00:05:30.810
The green number-- the
middle number-- comes from

00:05:30.810 --> 00:05:32.330
this middle a plus b.

00:05:32.330 --> 00:05:35.210
And then the blue number comes
from this right hand a plus b.

00:05:35.210 --> 00:05:37.080
And you saw me, I went all
the way through it, right?

00:05:37.080 --> 00:05:39.340
So hopefully you
believe this point.

00:05:39.340 --> 00:05:40.990
So let's think about it a
couple of different ways.

00:05:44.800 --> 00:05:48.550
To generate each of these terms
in the expansion of a plus b to

00:05:48.550 --> 00:05:54.360
the third, we're picking either
a or b from-- from the yellow a

00:05:54.360 --> 00:05:55.880
plus b, we're picking
either a or b.

00:05:55.880 --> 00:05:56.060
Right?

00:05:56.060 --> 00:05:57.630
We picked an a here,
we picked an a here.

00:05:57.630 --> 00:06:00.920
We picked a b here, a b here,
an a here, an a here, a b--.

00:06:00.920 --> 00:06:03.140
And the group from the green
a plus b, we're picking

00:06:03.140 --> 00:06:04.580
either an a or a b.

00:06:04.580 --> 00:06:07.760
And then from the blue a
plus b, we're picking

00:06:07.760 --> 00:06:09.090
either an a or a b.

00:06:09.090 --> 00:06:09.890
Right?

00:06:09.890 --> 00:06:13.450
So essentially the expansion,
if you think about it, this

00:06:13.450 --> 00:06:22.950
expansion-- we've essentially
done every way of choosing

00:06:22.950 --> 00:06:26.100
3 different things.

00:06:26.100 --> 00:06:28.910
Every way of picking either
an a or a b, from these

00:06:28.910 --> 00:06:30.130
3 different terms.

00:06:30.130 --> 00:06:32.700
And that ends up with
these-- what is this?

00:06:32.700 --> 00:06:37.140
1, 2, 3, 4, 5, 6, 7, 8 terms.

00:06:37.140 --> 00:06:38.640
Right?

00:06:38.640 --> 00:06:40.990
Now let's make it a little
bit-- let's give you a

00:06:40.990 --> 00:06:42.150
little bit more intuition
of what's going on.

00:06:42.150 --> 00:06:46.640
And I think if you're starting
to realize why this is dealing

00:06:46.640 --> 00:06:49.730
with permutations and
combinations, in particular.

00:06:49.730 --> 00:06:51.410
Once we simplify
it, what happens?

00:06:51.410 --> 00:06:52.990
This is a to the cubed, right?

00:06:52.990 --> 00:06:55.130
That's the only a cubed term.

00:06:55.130 --> 00:06:56.310
This is a squared b.

00:06:56.310 --> 00:06:58.920
What are the other
a squared b terms?

00:06:58.920 --> 00:07:00.440
Let's see, a squared b.

00:07:00.440 --> 00:07:01.760
This is also a squared b.

00:07:01.760 --> 00:07:03.860
So let me write down
all the a squared.

00:07:03.860 --> 00:07:05.800
So let's see-- let's
see how many a squared

00:07:05.800 --> 00:07:07.210
b terms there are.

00:07:07.210 --> 00:07:08.670
I'll do it in a neutral color.

00:07:08.670 --> 00:07:13.000
So this is a squared b.

00:07:13.000 --> 00:07:17.360
This is ba squared, but
that's also a squared b.

00:07:17.360 --> 00:07:18.490
What's another a squared b?

00:07:18.490 --> 00:07:21.810
This is also a squared b,
right? a times a times b.

00:07:21.810 --> 00:07:24.460
So there was 3 ways
to get a squared b.

00:07:24.460 --> 00:07:27.800
And that's why when we
eventually write the expansion,

00:07:27.800 --> 00:07:30.230
it's going to-- we know that
the coefficient in front of it

00:07:30.230 --> 00:07:33.840
is 3a squared b, right?

00:07:33.840 --> 00:07:35.880
The coefficient on the a
squared b term, when we

00:07:35.880 --> 00:07:36.620
actually multiply it out.

00:07:36.620 --> 00:07:39.870
And we've done that several
times already, when we did the

00:07:39.870 --> 00:07:43.360
binomial theorem, which we
took to the third power.

00:07:43.360 --> 00:07:46.740
So where did this 3 come from,
and why is that the same thing

00:07:46.740 --> 00:07:50.070
as when we learned the
definition of the

00:07:50.070 --> 00:07:50.730
binomial theorem?

00:07:50.730 --> 00:07:53.770
Why is, you know, does it just
happen to be the case that

00:07:53.770 --> 00:08:01.360
that's the same thing as 3
choose 2 times a squared b?

00:08:01.360 --> 00:08:01.880
Well, no.

00:08:01.880 --> 00:08:03.250
Think of it this way.

00:08:03.250 --> 00:08:06.090
We already know that every
term here, every term in the

00:08:06.090 --> 00:08:09.880
expansion, we're essentially
picking either an a or a b

00:08:09.880 --> 00:08:10.970
from each of these, right?

00:08:10.970 --> 00:08:13.500
We have to pick one term
from each of these.

00:08:13.500 --> 00:08:16.720
So the way you could think of
it, for a squared b-- to get a

00:08:16.720 --> 00:08:20.410
squared b-- we have to
essentially say, well,

00:08:20.410 --> 00:08:22.040
how many combinations?

00:08:22.040 --> 00:08:22.990
And that's the key word.

00:08:22.990 --> 00:08:26.440
How many combinations are
there, where out of these

00:08:26.440 --> 00:08:31.910
3 a plus b terms, I'm
choosing the a term?

00:08:31.910 --> 00:08:33.920
I'm choosing 2 a terms.

00:08:33.920 --> 00:08:38.590
Because to get a squared, I
have to pick an a term twice.

00:08:38.590 --> 00:08:41.940
And so that's where I have
to pick 2 a terms out of

00:08:41.940 --> 00:08:43.970
the three times I pick.

00:08:43.970 --> 00:08:45.630
So I'm picking three times.

00:08:45.630 --> 00:08:47.650
Two of the times, I'm
picking an a term.

00:08:47.650 --> 00:08:51.050
So out of three times,
I'm choosing 2.

00:08:51.050 --> 00:08:53.560
And that's where 3 choose
2 comes from, for the

00:08:53.560 --> 00:08:55.880
a squared b term.

00:08:55.880 --> 00:09:00.950
And so you could, for the ab
squared term, you could say,

00:09:00.950 --> 00:09:03.100
well, I'm picking an a once.

00:09:03.100 --> 00:09:06.780
How many ways are there to
choose a once when I'm

00:09:06.780 --> 00:09:08.120
taking from 3 things?

00:09:08.120 --> 00:09:12.130
So it could be 3 choose 1.

00:09:12.130 --> 00:09:16.200
But that's also the same thing,
or should be equal to-- you

00:09:16.200 --> 00:09:18.360
could also say, well, I'm
picking-- how many ways are

00:09:18.360 --> 00:09:20.770
there to pick b twice if
I'm picking three times?

00:09:20.770 --> 00:09:24.870
Well, if it's a b squared,
I'm picking b twice.

00:09:24.870 --> 00:09:29.110
So that should be equal to
3 choose 2 ab squared.

00:09:29.110 --> 00:09:31.960
And if you work these out, you
will find that, yes, these

00:09:31.960 --> 00:09:33.790
both turn out to be 3.

00:09:33.790 --> 00:09:37.480
And actually, that's why
there's some symmetry there,

00:09:37.480 --> 00:09:39.330
and the combinations
all work out.

00:09:39.330 --> 00:09:40.810
But hopefully, that's
giving you an intuition.

00:09:40.810 --> 00:09:44.340
Essentially, when you're
doing the-- so the binomial

00:09:44.340 --> 00:09:45.280
expansion of this--.

00:09:45.280 --> 00:09:48.870
Let me just rewrite it again.

00:09:48.870 --> 00:09:50.260
You know, this is a
plus b to the third.

00:09:50.260 --> 00:09:59.150
It's 3 choose 0 of a cubed b to
the zero, plus 3 choose 1 of

00:09:59.150 --> 00:10:03.530
a squared b to the
one, we could say.

00:10:03.530 --> 00:10:14.860
Plus 3 choose 2 of ab squared,
plus 3 choose 3 of a

00:10:14.860 --> 00:10:17.420
to the zero, b cubed.

00:10:17.420 --> 00:10:18.350
So what's this saying?

00:10:18.350 --> 00:10:20.710
What's 3 choose 3 saying?

00:10:20.710 --> 00:10:26.180
How many ways-- if I'm choosing
from 3 different things-- how

00:10:26.180 --> 00:10:29.670
many ways can I pick
exactly 3 b's?

00:10:29.670 --> 00:10:30.490
That's how you can view it.

00:10:30.490 --> 00:10:32.200
How many ways can I pick 3 b's?

00:10:32.200 --> 00:10:33.750
I'm either picking
an a or a b, right?

00:10:33.750 --> 00:10:35.660
You could say it's either a
heads or tails, or you know,

00:10:35.660 --> 00:10:38.570
red or black or white,
but it's either a or b.

00:10:38.570 --> 00:10:42.800
How many ways can I choose b
three times from 3 things?

00:10:42.800 --> 00:10:44.990
Well, when you evaluate this,
you get this to b one,

00:10:44.990 --> 00:10:45.830
and that makes sense.

00:10:45.830 --> 00:10:47.710
Because it's 1b cubed.

00:10:47.710 --> 00:10:52.090
And similarly, I mean, you
could view this as how many

00:10:52.090 --> 00:10:56.260
ways, when I'm picking
out of 3 things, can

00:10:56.260 --> 00:10:58.760
I pick exactly 0 b's?

00:10:58.760 --> 00:10:59.670
That's 3 choose [? b. ?]

00:10:59.670 --> 00:11:02.110
How many ways can I
pick exactly 0 b's?

00:11:02.110 --> 00:11:03.780
Well, that's the same thing
as how many ways can

00:11:03.780 --> 00:11:05.760
I pick exactly 3 a's?

00:11:05.760 --> 00:11:06.480
This is also 1.

00:11:06.480 --> 00:11:08.420
There's only 1 way to do it.

00:11:08.420 --> 00:11:09.520
And that's the way.

00:11:09.520 --> 00:11:11.710
Once again, there's only
1 way to do this one,

00:11:11.710 --> 00:11:13.380
and this was the way.

00:11:13.380 --> 00:11:16.220
There's 3 ways to
get a squared b.

00:11:16.220 --> 00:11:19.980
There's 3 combinations.

00:11:19.980 --> 00:11:24.260
There's 3 possible-- actually,
well, unique permutations.

00:11:24.260 --> 00:11:26.610
But they're all the
same combination.

00:11:26.610 --> 00:11:30.340
So there's 3 identical
combinations for picking 2 a's

00:11:30.340 --> 00:11:35.490
and a b, and those are this
one, this one, and this one.

00:11:35.490 --> 00:11:37.440
So hopefully, I
didn't confuse you.

00:11:37.440 --> 00:11:41.830
And hopefully at minimum, that
gives you a glancing intuition

00:11:41.830 --> 00:11:45.040
of why combinations are even
involved in the binomial

00:11:45.040 --> 00:11:47.180
theorem, or whether they're
even involved when you're

00:11:47.180 --> 00:11:49.400
expanding a binomial
to some power.

00:11:49.400 --> 00:11:53.020
And at best, I really hope that
I've given you a deep intuition

00:11:53.020 --> 00:11:54.430
for why this happens.

00:11:54.430 --> 00:11:57.770
I will see you in
the next video.


Proof: d/dx(x^n)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.000
I just did several videos on
the binomial theorem, so I

00:00:04.000 --> 00:00:07.550
think, now that they're done, I
think now is good time to do

00:00:07.550 --> 00:00:13.660
the proof of the derivative
of the general form.

00:00:13.660 --> 00:00:18.500
Let's take the derivative
of x to the n.

00:00:18.500 --> 00:00:20.300
Now that we know the
binomial theorem, we

00:00:20.300 --> 00:00:22.440
have the tools to do it.

00:00:22.440 --> 00:00:23.830
How do we take the derivative?

00:00:23.830 --> 00:00:26.070
Well, what's the classic
definition of the derivative?

00:00:26.070 --> 00:00:36.220
It is the limit as delta x
approaches zero of f of

00:00:36.220 --> 00:00:39.130
x plus delta x, right?

00:00:39.130 --> 00:00:44.210
So f of x plus delta x in this
situation is x plus delta

00:00:44.210 --> 00:00:48.000
x to the nth power, right?

00:00:48.000 --> 00:00:52.720
Minus f of x, well f of x
here is just x to the n.

00:00:52.720 --> 00:00:54.100
All of that over delta x.

00:00:57.710 --> 00:01:01.610
Now that we know the binomial
theorem we can figure out

00:01:01.610 --> 00:01:05.200
what the expansion of x plus
delta x is to the nth power.

00:01:05.200 --> 00:01:07.340
And if you don't know the
binomial theorem, go to my

00:01:07.340 --> 00:01:10.040
pre-calculus play list
and watch the videos on

00:01:10.040 --> 00:01:11.710
the binomial theorem.

00:01:11.710 --> 00:01:15.290
The binomial theorem tells us
that this is equal to-- I'm

00:01:15.290 --> 00:01:20.170
going to need some space for
this one-- the limit as

00:01:20.170 --> 00:01:23.310
delta x approaches zero.

00:01:23.310 --> 00:01:24.860
And what's the
binomial theorem?

00:01:24.860 --> 00:01:26.960
This is going to be equal to--
I'm just going to do the

00:01:26.960 --> 00:01:39.630
numerator-- x to the
n plus n choose 1.

00:01:39.630 --> 00:01:42.440
Once again, review the binomial
theorem if this is looks like

00:01:42.440 --> 00:01:44.340
latin to you and you
don't know latin.

00:01:44.340 --> 00:02:00.780
n choose 1 of x to the n minus
1 delta x plus n choose 2 x to

00:02:00.780 --> 00:02:07.650
the n minus 2, that's x n
minus 2, delta x squared.

00:02:07.650 --> 00:02:10.490
Then plus, and we have a bunch
of the digits, and in this

00:02:10.490 --> 00:02:12.560
proof we don't have to go
through all the digits but the

00:02:12.560 --> 00:02:14.810
binomial theorem tells us what
they are and, of course, the

00:02:14.810 --> 00:02:20.790
last digit we just keep adding
is going to be 1-- it would

00:02:20.790 --> 00:02:22.450
be n choose n which is 1.

00:02:22.450 --> 00:02:26.030
Let me just write that
down. n choose n.

00:02:26.030 --> 00:02:36.920
It's going to be x to the
zero times delta x to the n.

00:02:36.920 --> 00:02:39.030
So that's the
binomial expansion.

00:02:39.030 --> 00:02:44.450
Let me switch back to minus,
green that's x plus delta x

00:02:44.450 --> 00:02:49.740
to the n, so minus
x to the n power.

00:02:49.740 --> 00:02:51.840
That's x to the n, I know
I squashed it there.

00:02:51.840 --> 00:02:56.810
All of that over delta x.

00:02:56.810 --> 00:02:58.590
Let's see if we can simplify.

00:02:58.590 --> 00:03:00.940
First of all we have an x to
the n here, and at the very end

00:03:00.940 --> 00:03:05.880
we subtract out an x to the
n, so these two cancel out.

00:03:05.880 --> 00:03:09.080
If we look at every term here,
every term in the numerator has

00:03:09.080 --> 00:03:13.310
a delta x, so we can divide
the numerator and the

00:03:13.310 --> 00:03:16.820
denominator by delta x.

00:03:16.820 --> 00:03:19.210
This is the same thing
as 1 over delta x times

00:03:19.210 --> 00:03:19.960
this whole thing.

00:03:19.960 --> 00:03:29.670
So that is equal to the limit
as delta x approaches zero of,

00:03:29.670 --> 00:03:31.910
so we divide the top and the
bottom by delta x, or we

00:03:31.910 --> 00:03:34.530
multiply the numerator
times 1 over delta x.

00:03:34.530 --> 00:03:42.310
We get n choose 1 x
to the n minus 1.

00:03:42.310 --> 00:03:45.580
What's delta x divided by
delta x, that's just 1.

00:03:45.580 --> 00:03:52.440
Plus n choose 2, x
to the n minus 2.

00:03:52.440 --> 00:03:56.310
This is delta x squared, but
we divide by delta x we

00:03:56.310 --> 00:03:57.940
just get a delta x here.

00:03:57.940 --> 00:03:59.490
Delta x.

00:03:59.490 --> 00:04:02.100
And then we keep having a bunch
of terms, we're going to divide

00:04:02.100 --> 00:04:04.180
all of them by delta x.

00:04:04.180 --> 00:04:07.190
And then the last term is
delta x to the n, but then

00:04:07.190 --> 00:04:09.130
we're going to divide
that by delta x.

00:04:09.130 --> 00:04:14.550
So the last term becomes n
choose n, x to the zero is 1,

00:04:14.550 --> 00:04:17.940
we can ignore that. delta x
to the n divided by delta x.

00:04:17.940 --> 00:04:21.910
Well that's delta x
to the n minus 1.

00:04:25.150 --> 00:04:26.690
Then what are we doing now?

00:04:26.690 --> 00:04:28.505
Remember, we're taking
the limit as delta

00:04:28.505 --> 00:04:30.760
x approaches zero.

00:04:30.760 --> 00:04:33.220
As delta x approaches zero,
pretty much every term that

00:04:33.220 --> 00:04:37.290
has a delta x in it,
it becomes zero.

00:04:37.290 --> 00:04:40.850
When you multiply but
zero, you get zero.

00:04:40.850 --> 00:04:42.650
This first term has no
delta x in it, but

00:04:42.650 --> 00:04:43.730
every other term does.

00:04:43.730 --> 00:04:45.950
Every other term, even after
we divided by delta x

00:04:45.950 --> 00:04:46.680
has a delta x in it.

00:04:46.680 --> 00:04:47.930
So that's a zero.

00:04:47.930 --> 00:04:50.970
Every term is zero, all
of the other n minus 1

00:04:50.970 --> 00:04:52.940
terms, they're all zeros.

00:04:52.940 --> 00:04:58.390
All we're left with is that
this is equal to n choose

00:04:58.390 --> 00:05:02.520
1 of x the n minus 1.

00:05:02.520 --> 00:05:04.530
And what's n choose 1?

00:05:04.530 --> 00:05:13.070
That equals n factorial over 1
factorial divided by n minus 1

00:05:13.070 --> 00:05:17.170
factorial times x
to the n minus 1.

00:05:17.170 --> 00:05:19.390
1 factorial is 1.

00:05:19.390 --> 00:05:23.430
If I have 7 factorial divided
by 6 factorial, that's just 1.

00:05:23.430 --> 00:05:26.070
Or if I have 3 factorial
divided by 2 factorial, that's

00:05:26.070 --> 00:05:27.650
just 3, you can work it out.

00:05:27.650 --> 00:05:30.660
10 factorial divided by
9 factorial that's 10.

00:05:30.660 --> 00:05:32.830
So n factorial divided
by n minus 1 factorial,

00:05:32.830 --> 00:05:34.600
that's just equal to n.

00:05:34.600 --> 00:05:39.540
So this is equal to n
times x to the n minus 1.

00:05:39.540 --> 00:05:43.866
That's the derivative
of x to the n. n times

00:05:43.866 --> 00:05:45.900
x to the n minus 1.

00:05:45.900 --> 00:05:52.470
We just proved the derivative
for any positive integer when

00:05:52.470 --> 00:05:55.610
x to the power n, where n
is any positive integer.

00:05:55.610 --> 00:05:59.790
And we see later it actually
works for all real

00:05:59.790 --> 00:06:02.990
numbers and the exponent.

00:06:02.990 --> 00:06:05.980
I will see you in
a future video.

00:06:05.980 --> 00:06:08.860
Another thing I wanted to point
out is, you know I said that

00:06:08.860 --> 00:06:10.150
we had to know the
binomial theorem.

00:06:10.150 --> 00:06:12.050
But if you think about it, we
really didn't even have to know

00:06:12.050 --> 00:06:15.630
the binomial theorem because we
knew in any binomial

00:06:15.630 --> 00:06:17.960
expansion-- I mean, you'd have
to know a little bit-- but if

00:06:17.960 --> 00:06:19.350
you did a little
experimentation you would

00:06:19.350 --> 00:06:25.470
realize that whenever you
expand a plus b to the nth

00:06:25.470 --> 00:06:29.040
power, first term is going to
be a to the n, and the second

00:06:29.040 --> 00:06:35.670
term is going to be plus n a to
the n minus 1 b.

00:06:35.670 --> 00:06:37.320
And then you are going to keep
having a bunch of terms.

00:06:37.320 --> 00:06:40.190
But these are the only terms
that are relevant to this proof

00:06:40.190 --> 00:06:43.575
because all the other terms get
canceled out when delta

00:06:43.575 --> 00:06:44.980
x approaches zero.

00:06:44.980 --> 00:06:47.110
So if you just knew that you
could have done this, but it's

00:06:47.110 --> 00:06:49.150
much better to do it with
the binomial theorem.

00:06:49.150 --> 00:06:50.880
Ignore what I just said
if it confused you.

00:06:50.880 --> 00:06:55.280
I'm just saying that we could
have just said the rest of

00:06:55.280 --> 00:06:56.950
these terms all go to zero.

00:06:56.950 --> 00:06:59.700
Anyway, hopefully you
found that fulfilling.

00:06:59.700 --> 00:07:01.960
I will see you in
future videos.