Decomposing shapes to find area (add)

WEBVTT
Kind: captions
Language: en

00:00:01.259 --> 00:00:03.596
- [Voiceover] What is
the area of the figure?

00:00:03.596 --> 00:00:06.730
So down here we have this one, two, three,

00:00:06.730 --> 00:00:11.656
four, five, six, seven,
eight, nine, 10-sided figure,

00:00:11.656 --> 00:00:13.173
and we want to know its area,

00:00:13.173 --> 00:00:17.656
how many square meters
does this figure cover?

00:00:17.656 --> 00:00:18.672
And we have some measurements,

00:00:18.672 --> 00:00:20.442
that seems helpful,

00:00:20.442 --> 00:00:21.807
but what's not too helpful to me is

00:00:21.807 --> 00:00:23.090
I don't know the special trick

00:00:23.090 --> 00:00:26.173
to find the area of a 10-sided figure

00:00:27.058 --> 00:00:29.148
so I've got to think about what I do know

00:00:29.148 --> 00:00:30.280
and what I do know

00:00:30.280 --> 00:00:33.153
is the way to find area of a rectangle.

00:00:33.153 --> 00:00:36.375
So what I can do, because I can see,

00:00:36.375 --> 00:00:38.380
if I can find any rectangles in here.

00:00:38.380 --> 00:00:41.952
Here's one rectangle, right there.

00:00:41.952 --> 00:00:44.755
So I can find the area of that part.

00:00:44.755 --> 00:00:46.403
Then let's see if I can find any more.

00:00:46.403 --> 00:00:48.486
Here's another rectangle.

00:00:51.863 --> 00:00:54.631
So I can find the area of that part.

00:00:54.631 --> 00:00:57.359
We could call that one
a rectangle or a square.

00:00:57.359 --> 00:01:01.421
And then that leaves
us with this last part,

00:01:01.421 --> 00:01:03.754
which is again, a rectangle.

00:01:05.852 --> 00:01:07.739
So what we did is, we broke this up

00:01:07.739 --> 00:01:11.077
or decomposed it into three rectangles

00:01:11.077 --> 00:01:12.644
and now if I find out how much space

00:01:12.644 --> 00:01:14.617
this purple one covers,

00:01:14.617 --> 00:01:17.258
and the blue one and the green one,

00:01:17.258 --> 00:01:19.783
if I combine those, that
would tell me the area

00:01:19.783 --> 00:01:21.148
of the entire figure,

00:01:21.148 --> 00:01:24.283
how much space the entire figure covers.

00:01:24.283 --> 00:01:25.938
So let's start with this one right here.

00:01:25.938 --> 00:01:28.433
This one is three meters long,

00:01:28.433 --> 00:01:30.771
so we can kind of divide
that by three meters,

00:01:30.771 --> 00:01:33.142
into three equal meters,

00:01:33.142 --> 00:01:35.929
and then we've got a width
of two meters down here

00:01:35.929 --> 00:01:37.960
so we can split that in half.

00:01:37.960 --> 00:01:40.460
So if we draw those lines out,

00:01:42.488 --> 00:01:46.312
we can see this top row is
going to cover one square meter,

00:01:46.312 --> 00:01:49.505
two square meters, three square meters,

00:01:49.505 --> 00:01:51.072
and then there's two rows of that,

00:01:51.072 --> 00:01:52.655
so there's two rows

00:01:53.696 --> 00:01:55.690
of three square meters

00:01:55.690 --> 00:01:59.169
for a total of six square meters.

00:01:59.169 --> 00:02:02.174
This rectangle covers six square meters,

00:02:02.174 --> 00:02:04.507
so this part of the entire figure

00:02:04.507 --> 00:02:06.915
covers six square meters.

00:02:06.915 --> 00:02:10.863
The next one, our measurements
are three and a three,

00:02:10.863 --> 00:02:13.765
so it will have three rows
of three square meters

00:02:13.765 --> 00:02:15.598
or nine square meters,

00:02:17.093 --> 00:02:19.377
and then finally this purple one

00:02:19.377 --> 00:02:21.205
has three meters and nine,

00:02:21.205 --> 00:02:24.112
so we can say it will
have three rows of nine

00:02:24.112 --> 00:02:25.965
or nine rows of three square meters

00:02:25.965 --> 00:02:28.132
which is 27 square meters.

00:02:29.685 --> 00:02:30.602
So the area

00:02:32.036 --> 00:02:34.648
of this purple section,

00:02:34.648 --> 00:02:38.073
it covers completely 27 square meters.

00:02:38.073 --> 00:02:40.772
The green covers nine square meters,

00:02:40.772 --> 00:02:43.907
and the blue covered six square meters.

00:02:43.907 --> 00:02:47.012
So, if we combine all those areas,

00:02:47.012 --> 00:02:48.678
all those square meters it covers,

00:02:48.678 --> 00:02:51.989
that will tell us the
area of the entire figure.

00:02:51.989 --> 00:02:54.457
So we have six square meters,

00:02:54.457 --> 00:02:57.098
plus nine square meters,

00:02:57.098 --> 00:02:57.931
plus 27,

00:02:59.584 --> 00:03:03.546
and we can solve that,
six plus nine is 15,

00:03:03.546 --> 00:03:05.379
15 plus 27, let's see,

00:03:06.492 --> 00:03:08.603
five ones and seven ones

00:03:08.603 --> 00:03:09.520
is 12 ones.

00:03:11.713 --> 00:03:13.455
We'll just find some space up there.

00:03:13.455 --> 00:03:14.288
And

00:03:15.130 --> 00:03:16.676
one 10 and two 10's

00:03:16.676 --> 00:03:18.402
or a 10 and a 20

00:03:18.402 --> 00:03:19.235
is 30.

00:03:21.041 --> 00:03:22.791
And 30 plus 12 is 42.

00:03:24.350 --> 00:03:26.701
So the area of the entire figure is

00:03:26.701 --> 00:03:27.534
42

00:03:28.656 --> 00:03:29.489
square

00:03:31.271 --> 00:03:32.104
meters.


Decomposing shapes to find area (grids)

WEBVTT
Kind: captions
Language: en

00:00:01.541 --> 00:00:03.495
- [Voiceover] Each small
square in the diagram

00:00:03.495 --> 00:00:06.495
has a side length of one centimeter.

00:00:07.533 --> 00:00:11.243
So, what is the area of the figure?

00:00:11.243 --> 00:00:13.798
So, we have this figure down here in blue,

00:00:13.798 --> 00:00:15.395
and we want to know its area.

00:00:15.395 --> 00:00:18.228
Area is the total space it covers.

00:00:19.321 --> 00:00:22.545
And, we're also told that
each of these little squares

00:00:22.545 --> 00:00:25.450
has a side length of one centimeter.

00:00:25.450 --> 00:00:27.232
So, that means that each of these squares

00:00:27.232 --> 00:00:29.315
is one square centimeter.

00:00:30.744 --> 00:00:32.476
So, we can find the area by seeing

00:00:32.476 --> 00:00:37.250
how many square centimeters
does this figure cover?

00:00:37.250 --> 00:00:39.974
One way would be to just try

00:00:39.974 --> 00:00:43.375
to draw the little square
centimeters and count them.

00:00:43.375 --> 00:00:45.543
There's one square
centimeter, there's two,

00:00:45.543 --> 00:00:47.519
and so on and keep counting them

00:00:47.519 --> 00:00:49.843
all the way through.

00:00:49.843 --> 00:00:52.040
Or, what we could do

00:00:52.040 --> 00:00:53.208
is we could look at this

00:00:53.208 --> 00:00:56.208
and try to break it into two shapes.

00:00:57.555 --> 00:00:59.555
So we can say down here,

00:01:00.483 --> 00:01:01.683
into two rectangles.

00:01:01.683 --> 00:01:04.947
Down here we have one rectangle,

00:01:04.947 --> 00:01:08.197
and up here we have a second rectangle.

00:01:10.743 --> 00:01:13.548
And then we can find the
area of each rectangle

00:01:13.548 --> 00:01:17.019
and add it together to find the total area

00:01:17.019 --> 00:01:18.936
that the figure covers.

00:01:19.979 --> 00:01:23.396
Down here on the bottom, we have two rows

00:01:24.331 --> 00:01:25.286
of unit squares.

00:01:25.286 --> 00:01:27.757
And each of those has one, two,

00:01:27.757 --> 00:01:30.257
three, four, five, six, seven.

00:01:31.536 --> 00:01:34.403
So, one, two, three,
four, five, six, seven.

00:01:34.403 --> 00:01:37.903
So there's two rows of seven unit squares,

00:01:39.268 --> 00:01:40.826
or seven square centimeters,

00:01:40.826 --> 00:01:44.993
so the bottom rectangle is made
up of 14 square centimeters.

00:01:47.259 --> 00:01:49.842
It covers 14 square centimeters.

00:01:49.842 --> 00:01:51.864
And the top rectangle,

00:01:51.864 --> 00:01:56.031
let's see we have one row,
two, three, four, five rows.

00:01:59.378 --> 00:02:03.135
And each of those rows has
one, two square centimeters,

00:02:03.135 --> 00:02:07.302
so we have five rows of two
square centimeters, or 10.

00:02:10.170 --> 00:02:13.761
So, this top rectangle
here that we have in blue

00:02:13.761 --> 00:02:16.518
covers 10 square centimeters,

00:02:16.518 --> 00:02:20.685
plus the bottom rectangle
that we outlined in green

00:02:21.931 --> 00:02:24.348
covers 14 square centimeters,

00:02:25.402 --> 00:02:26.402
so in total,

00:02:27.699 --> 00:02:29.949
the entire figure covers 24

00:02:31.315 --> 00:02:32.148
square

00:02:34.248 --> 00:02:35.248
centimeters.

00:02:36.103 --> 00:02:39.916
So, 24 square centimeters is our area,

00:02:39.916 --> 00:02:43.892
because area is how much
space does it cover,

00:02:43.892 --> 00:02:48.059
and we figured out that it
covered 24 square centimeters.


Harmonic Functions

WEBVTT
Kind: captions
Language: en

00:00:00.385 --> 00:00:01.284
- [Voiceover] Hello everyone.

00:00:01.284 --> 00:00:04.458
So here, I'd like to talk
about harmonic functions.

00:00:04.458 --> 00:00:05.803
Now, harmonic functions

00:00:05.803 --> 00:00:06.754
are a very special kind

00:00:06.754 --> 00:00:08.864
of multivariable function,

00:00:08.864 --> 00:00:10.710
and they're defined in
terms of the Laplacian,

00:00:10.710 --> 00:00:11.610
which I've been talking about

00:00:11.610 --> 00:00:12.969
in the last few videos.

00:00:12.969 --> 00:00:14.470
So the Laplacian, which we denote

00:00:14.470 --> 00:00:18.104
with this upper right-side-up triangle,

00:00:18.104 --> 00:00:20.110
is an operator that you might take

00:00:20.110 --> 00:00:21.625
on a multivariable function.

00:00:21.625 --> 00:00:22.693
So it might have two inputs,

00:00:22.693 --> 00:00:24.573
it could have, you know, a hundred inputs,

00:00:24.573 --> 00:00:26.350
just some kind of multivariable function

00:00:26.350 --> 00:00:27.906
with a scalar output.

00:00:27.906 --> 00:00:29.447
And I talked about it
in the last few videos,

00:00:29.447 --> 00:00:30.280
but as a reminder,

00:00:30.280 --> 00:00:31.463
it's defined as the divergence

00:00:31.463 --> 00:00:33.213
of the gradient of F,

00:00:34.346 --> 00:00:36.631
and it's kind of like
the second derivative.

00:00:36.631 --> 00:00:38.223
It's sort of the way to extend the idea

00:00:38.223 --> 00:00:39.581
of the second derivative

00:00:39.581 --> 00:00:41.282
into multiple dimensions.

00:00:41.282 --> 00:00:43.102
Now, what a harmonic function is,

00:00:43.102 --> 00:00:46.558
is one where the Laplacian
is equal to zero.

00:00:46.558 --> 00:00:47.391
And it's equal to zero

00:00:47.391 --> 00:00:49.328
at every possible input point.

00:00:49.328 --> 00:00:50.989
And sometimes the way that people

00:00:50.989 --> 00:00:52.311
write this to distinguish it,

00:00:52.311 --> 00:00:54.414
they'll make a kind of triple equals sign,

00:00:54.414 --> 00:00:56.235
maybe saying like, "Equivalent to zero."

00:00:56.235 --> 00:00:58.300
And this is really just
a way of emphasizing

00:00:58.300 --> 00:00:59.719
that it's equal to zero

00:00:59.719 --> 00:01:00.915
at all possible input points.

00:01:00.915 --> 00:01:02.352
It's, you know, not an equation

00:01:02.352 --> 00:01:04.149
that you're solving for
the specific X and Y,

00:01:04.149 --> 00:01:05.313
where it equals zero.

00:01:05.313 --> 00:01:07.078
It's a statement about the function.

00:01:07.078 --> 00:01:08.682
And to get our head around this,

00:01:08.682 --> 00:01:10.078
because it's kind of a,

00:01:10.078 --> 00:01:10.932
you know, as you're just starting

00:01:10.932 --> 00:01:12.517
to learn about the Laplacian,

00:01:12.517 --> 00:01:13.359
it's hard to just immediately

00:01:13.359 --> 00:01:15.754
see the intuition for what this means,

00:01:15.754 --> 00:01:16.587
let's think about what it means

00:01:16.587 --> 00:01:18.338
for a single variable function.

00:01:18.338 --> 00:01:21.474
If you just have some single
variable function of X,

00:01:21.474 --> 00:01:22.962
and you're looking at
its second derivative,

00:01:22.962 --> 00:01:25.417
which is kind of the
analog of the Laplacian,

00:01:25.417 --> 00:01:27.498
what does it mean if that's equal to zero?

00:01:27.498 --> 00:01:29.094
Well, we can integrate it,

00:01:29.094 --> 00:01:30.661
we can take the antiderivative,

00:01:30.661 --> 00:01:34.526
and say, that means that
the single derivative of F,

00:01:34.526 --> 00:01:35.383
well let's see, what functions

00:01:35.383 --> 00:01:37.372
have a derivative that's zero?

00:01:37.372 --> 00:01:39.173
The only functions are the constant ones,

00:01:39.173 --> 00:01:41.564
so C is just gonna mean
some constant here.

00:01:41.564 --> 00:01:42.808
And if you integrate that again,

00:01:42.808 --> 00:01:44.164
say what function has,

00:01:44.164 --> 00:01:46.241
as its derivative, a constant,

00:01:46.241 --> 00:01:48.743
well, it's gonna be that constant times X,

00:01:48.743 --> 00:01:50.097
plus some other constant,

00:01:50.097 --> 00:01:51.985
some other constant K.

00:01:51.985 --> 00:01:53.663
So basically, linear functions.

00:01:53.663 --> 00:01:54.924
So if you're thinking of a graph,

00:01:54.924 --> 00:01:56.972
it's just something that's got a line,

00:01:56.972 --> 00:01:58.566
line passing through it like that.

00:01:58.566 --> 00:02:00.116
And this should kind of make sense

00:02:00.116 --> 00:02:01.872
if you think of the
geometric interpretation

00:02:01.872 --> 00:02:03.500
for the second derivative.

00:02:03.500 --> 00:02:05.658
Because if you're just
looking at a random,

00:02:05.658 --> 00:02:06.851
you know, arbitrary function

00:02:06.851 --> 00:02:08.468
that's kind of curving as it does,

00:02:08.468 --> 00:02:11.128
the second derivative is negative

00:02:11.128 --> 00:02:12.762
when this concavity is down.

00:02:12.762 --> 00:02:14.374
So this right here would be a point

00:02:14.374 --> 00:02:15.391
where the second derivative,

00:02:15.391 --> 00:02:16.950
it's not zero, it's negative.

00:02:16.950 --> 00:02:18.823
And over here, when the concavity is up,

00:02:18.823 --> 00:02:20.780
and it's got a sort of bowl shape,

00:02:20.780 --> 00:02:24.177
that's where the second
derivative is positive.

00:02:24.177 --> 00:02:25.509
So if we're saying that
the second derivative

00:02:25.509 --> 00:02:27.063
has to always be zero,

00:02:27.063 --> 00:02:29.437
then it can't curve down
and it can't curve up,

00:02:29.437 --> 00:02:30.997
and it can't do that anywhere,

00:02:30.997 --> 00:02:33.023
so basically there's no curving allowed,

00:02:33.023 --> 00:02:35.294
so whatever direction it starts at,

00:02:35.294 --> 00:02:36.456
it's not allowed to curve,

00:02:36.456 --> 00:02:39.295
so it just sticks straight like that.

00:02:39.295 --> 00:02:40.850
But once we extend this to the idea

00:02:40.850 --> 00:02:42.567
of a multivariable function,

00:02:42.567 --> 00:02:43.901
things can get a lot more interesting

00:02:43.901 --> 00:02:45.443
than just a straight line.

00:02:45.443 --> 00:02:46.428
So an example,

00:02:46.428 --> 00:02:47.660
I've got the graph here

00:02:47.660 --> 00:02:49.886
of a multivariable function

00:02:49.886 --> 00:02:51.720
that happens to be harmonic.

00:02:51.720 --> 00:02:52.813
So the graph that you're looking at,

00:02:52.813 --> 00:02:54.818
this is of a two variable function,

00:02:54.818 --> 00:02:58.248
and the function specifically is

00:02:58.248 --> 00:02:59.081
F of XY,

00:03:00.256 --> 00:03:01.622
XY,

00:03:01.622 --> 00:03:04.001
is equal to E to the X

00:03:04.001 --> 00:03:05.362
multiplied by

00:03:05.362 --> 00:03:06.195
sine of Y.

00:03:08.080 --> 00:03:10.210
And as we're looking at the graph here,

00:03:10.210 --> 00:03:11.427
hopefully it makes a little bit of sense

00:03:11.427 --> 00:03:13.033
why this is sort of an E to the X

00:03:13.033 --> 00:03:14.790
sine of Y pattern.

00:03:14.790 --> 00:03:16.837
'Cause as we're moving in
the positive X direction,

00:03:16.837 --> 00:03:18.485
this here is the positive X direction,

00:03:18.485 --> 00:03:20.284
you have this exponential shape,

00:03:20.284 --> 00:03:22.245
and this corresponds with
the fact that over here

00:03:22.245 --> 00:03:23.857
we've got an E to the X,

00:03:23.857 --> 00:03:24.718
so as you move X,

00:03:24.718 --> 00:03:26.170
it kinda looks like E to the X,

00:03:26.170 --> 00:03:27.688
and it's being multiplied by something

00:03:27.688 --> 00:03:29.052
that is a function of Y.

00:03:29.052 --> 00:03:30.302
So if you're holding Y constant,

00:03:30.302 --> 00:03:31.858
this just looks like a constant.

00:03:31.858 --> 00:03:32.813
But notice, if that was a,

00:03:32.813 --> 00:03:33.952
if that was a negative constant,

00:03:33.952 --> 00:03:35.279
if sine of Y at some point

00:03:35.279 --> 00:03:36.631
happens to be negative,

00:03:36.631 --> 00:03:38.548
then your whole exponential function

00:03:38.548 --> 00:03:39.796
actually kind of goes down.

00:03:39.796 --> 00:03:41.919
It's sort of like a
negative E to the X look.

00:03:41.919 --> 00:03:44.088
But if you imagine moving
in the Y direction,

00:03:44.088 --> 00:03:46.803
so instead of the pure
X direction like that,

00:03:46.803 --> 00:03:49.669
if we imagine ourselves moving

00:03:49.669 --> 00:03:51.155
with the input going along,

00:03:51.155 --> 00:03:51.988
let's see what it would be,

00:03:51.988 --> 00:03:53.110
it would be this way,

00:03:53.110 --> 00:03:54.814
positive Y direction,

00:03:54.814 --> 00:03:57.120
you have this sort of sinusoidal shape,

00:03:57.120 --> 00:03:58.249
and that should make sense

00:03:58.249 --> 00:03:59.635
because you've got this sine of Y.

00:03:59.635 --> 00:04:01.401
And depending on what E to the X is,

00:04:01.401 --> 00:04:03.201
the amplitude of that sine wave

00:04:03.201 --> 00:04:04.059
is gonna get, you know,

00:04:04.059 --> 00:04:05.339
really high at some points here.

00:04:05.339 --> 00:04:07.016
It's going way up and way down.

00:04:07.016 --> 00:04:09.249
But if E to the X was
really small, you know,

00:04:09.249 --> 00:04:10.579
it hardly, hardly even looks

00:04:10.579 --> 00:04:11.703
like it's wiggling over here.

00:04:11.703 --> 00:04:13.916
It pretty much looks flat.

00:04:13.916 --> 00:04:15.824
So that's the graph that we're looking at.

00:04:15.824 --> 00:04:16.657
And I'm telling you right now,

00:04:16.657 --> 00:04:18.105
I claim that this is harmonic.

00:04:18.105 --> 00:04:19.882
This is a function whose Laplacian

00:04:19.882 --> 00:04:21.207
is equal to zero.

00:04:21.207 --> 00:04:23.039
And what that would mean is that

00:04:23.039 --> 00:04:24.790
as we go over here and we say,

00:04:24.790 --> 00:04:27.839
we evaluate the Laplacian of F,

00:04:27.839 --> 00:04:29.110
which, just to remind you,

00:04:29.110 --> 00:04:30.115
there's a different formula

00:04:30.115 --> 00:04:31.984
rather than thinking
divergence of gradient,

00:04:31.984 --> 00:04:33.685
that turns out to be completely the same

00:04:33.685 --> 00:04:35.845
as saying, you take the second derivative

00:04:35.845 --> 00:04:38.434
of that function with respect to X,

00:04:38.434 --> 00:04:39.859
that's its first input,

00:04:39.859 --> 00:04:41.324
and you add that,

00:04:41.324 --> 00:04:42.850
let's see, second derivative
with respect to X,

00:04:42.850 --> 00:04:45.111
you add that to the second derivative

00:04:45.111 --> 00:04:46.288
of your function

00:04:46.288 --> 00:04:47.825
with respect to the next variable.

00:04:47.825 --> 00:04:48.658
And you keep doing this

00:04:48.658 --> 00:04:49.995
for all of the different
variables that there are,

00:04:49.995 --> 00:04:51.898
but this is just a two variable function,

00:04:51.898 --> 00:04:53.253
so you do this twice.

00:04:53.253 --> 00:04:54.652
The claim is that this

00:04:54.652 --> 00:04:56.010
is always equal to zero.

00:04:56.010 --> 00:04:57.352
So I might say, kind of equivalent,

00:04:57.352 --> 00:04:58.867
at every possible input,

00:04:58.867 --> 00:05:00.733
it's equal to zero.

00:05:00.733 --> 00:05:01.566
And I think I'll leave that

00:05:01.566 --> 00:05:03.134
as something for you to compute.

00:05:03.134 --> 00:05:04.255
It might be kind of good practice

00:05:04.255 --> 00:05:05.960
to kind of get a feel for

00:05:05.960 --> 00:05:07.423
computing the Laplacian.

00:05:07.423 --> 00:05:08.587
But what I wanna do is interpret

00:05:08.587 --> 00:05:09.992
what does this actually mean, right?

00:05:09.992 --> 00:05:11.068
'Cause you can plug it through,

00:05:11.068 --> 00:05:11.901
and you can see, ah yes,

00:05:11.901 --> 00:05:13.801
at all possible inputs, it will be zero.

00:05:13.801 --> 00:05:14.920
But what does that mean?

00:05:14.920 --> 00:05:17.004
Because in the single variable context,

00:05:17.004 --> 00:05:18.740
once we started thinking about

00:05:18.740 --> 00:05:20.569
the geometric interpretation

00:05:20.569 --> 00:05:22.882
of a second derivative as this concavity,

00:05:22.882 --> 00:05:23.871
it sort of made sense

00:05:23.871 --> 00:05:25.033
that forcing it to be zero

00:05:25.033 --> 00:05:26.514
will give us a straight line.

00:05:26.514 --> 00:05:27.570
But clearly, that's not the case.

00:05:27.570 --> 00:05:28.701
This is much more complicated

00:05:28.701 --> 00:05:30.269
than a straight line.

00:05:30.269 --> 00:05:31.102
And for that,

00:05:31.102 --> 00:05:32.651
I want to give a kind of a different

00:05:32.651 --> 00:05:33.933
way that you can think about

00:05:33.933 --> 00:05:36.235
the single variable second derivative.

00:05:36.235 --> 00:05:37.476
On the one hand, you can think of,

00:05:37.476 --> 00:05:39.553
let's say, negative second derivative,

00:05:39.553 --> 00:05:40.752
as being this concavity

00:05:40.752 --> 00:05:42.467
where it's kind of frowning down.

00:05:42.467 --> 00:05:44.279
But another way you can
maybe think about this

00:05:44.279 --> 00:05:46.586
is saying that all of the
neighbors of your point,

00:05:46.586 --> 00:05:48.339
if you go a little bit to the left, right,

00:05:48.339 --> 00:05:49.736
you've got an input point here,

00:05:49.736 --> 00:05:51.195
and if you go a little bit to the left,

00:05:51.195 --> 00:05:52.872
the neighbor is less than it,

00:05:52.872 --> 00:05:54.168
and if you go a little bit to the right,

00:05:54.168 --> 00:05:56.126
that other neighbor is also less than it.

00:05:56.126 --> 00:05:57.917
So it's kind of a way of saying,

00:05:57.917 --> 00:06:00.089
hey, if you look at the
neighbors of your input,

00:06:00.089 --> 00:06:01.926
so if you, if you happen
to be making the claim

00:06:01.926 --> 00:06:04.906
that F double prime, at
some particular input,

00:06:04.906 --> 00:06:06.192
like X sub O,

00:06:06.192 --> 00:06:07.766
is less than zero,

00:06:07.766 --> 00:06:09.281
it's saying that all of the neighbors

00:06:09.281 --> 00:06:10.349
of X sub O,

00:06:10.349 --> 00:06:11.631
all of the neighbors of that point,

00:06:11.631 --> 00:06:13.022
are less than it.

00:06:13.022 --> 00:06:13.974
And if you do a similar thing

00:06:13.974 --> 00:06:15.873
over at a positive concavity point

00:06:15.873 --> 00:06:17.536
where it's kind of smiling up,

00:06:17.536 --> 00:06:19.286
you say, well its neighbor to the right

00:06:19.286 --> 00:06:20.200
has a greater value,

00:06:20.200 --> 00:06:22.376
and its neighbor to the
left has a greater value.

00:06:22.376 --> 00:06:25.035
So at some point, where
the second derivative,

00:06:25.035 --> 00:06:27.004
instead of being less than zero,

00:06:27.004 --> 00:06:28.171
happens to be,

00:06:29.160 --> 00:06:30.826
happens to be greater than zero,

00:06:30.826 --> 00:06:32.162
that means that the neighbors

00:06:32.162 --> 00:06:34.376
tend to be greater than the point itself.

00:06:34.376 --> 00:06:35.946
And even if you're
looking at a circumstance

00:06:35.946 --> 00:06:37.572
that isn't this idealized, you know,

00:06:37.572 --> 00:06:39.340
it happens to be a local minimum,

00:06:39.340 --> 00:06:42.723
but let's say you're looking at a graph.

00:06:42.723 --> 00:06:44.199
Let's say you're looking at a function

00:06:44.199 --> 00:06:46.248
at a point where it's concave up, right?

00:06:46.248 --> 00:06:49.710
It's concave up, but
it's not this idealized,

00:06:49.710 --> 00:06:51.388
local minimum kind of circumstance.

00:06:51.388 --> 00:06:52.371
So instead, you might be looking

00:06:52.371 --> 00:06:53.401
at a point like this,

00:06:53.401 --> 00:06:54.284
and if you look at, you know,

00:06:54.284 --> 00:06:55.666
its neighbor to the left,

00:06:55.666 --> 00:06:57.545
that'll have some value
that's actually less

00:06:57.545 --> 00:06:58.697
than your original guy,

00:06:58.697 --> 00:06:59.611
so the neighbor looks like

00:06:59.611 --> 00:07:01.044
it's less than it on the left,

00:07:01.044 --> 00:07:03.151
but if you move that same
distance to the right,

00:07:03.151 --> 00:07:05.076
its neighbor is greater.

00:07:05.076 --> 00:07:06.398
But you would say, on average,

00:07:06.398 --> 00:07:08.669
if you took the average
value of the neighbors,

00:07:08.669 --> 00:07:10.565
the neighbor on the
right kind of outbalances

00:07:10.565 --> 00:07:11.672
the neighbor on the left,

00:07:11.672 --> 00:07:13.022
and you would say, on average,

00:07:13.022 --> 00:07:15.815
its neighbors are greater
than the point itself.

00:07:15.815 --> 00:07:17.445
So let's say that input point there

00:07:17.445 --> 00:07:18.679
was like X sub O,

00:07:18.679 --> 00:07:20.476
that would mean that the second derivative

00:07:20.476 --> 00:07:22.926
of your function at that point,

00:07:22.926 --> 00:07:24.899
you know, is greater than zero.

00:07:24.899 --> 00:07:26.236
So this positive concavity,

00:07:26.236 --> 00:07:27.185
you can also think of it

00:07:27.185 --> 00:07:29.009
as a measure of, on average,

00:07:29.009 --> 00:07:31.206
are the neighbors greater
than your original point

00:07:31.206 --> 00:07:32.350
or less than it?

00:07:32.350 --> 00:07:33.205
And the reason I'm saying this

00:07:33.205 --> 00:07:35.186
is because this idea,

00:07:35.186 --> 00:07:36.583
of kind of comparing your neighbors

00:07:36.583 --> 00:07:37.705
to the original point

00:07:37.705 --> 00:07:39.958
is a much better way to contemplate

00:07:39.958 --> 00:07:42.360
the Laplacian in the multivariable world.

00:07:42.360 --> 00:07:43.833
So if we look at a function like this,

00:07:43.833 --> 00:07:45.114
and let's say we're looking at it

00:07:45.114 --> 00:07:47.687
kind of from a bird's eye view.

00:07:47.687 --> 00:07:49.771
So we've got our XY plane, right?

00:07:49.771 --> 00:07:51.632
This over here is the X-axis,

00:07:51.632 --> 00:07:54.012
and this up here is the Y-axis.

00:07:54.012 --> 00:07:54.845
And let's say that we're looking

00:07:54.845 --> 00:07:57.253
at some specific input point.

00:07:57.253 --> 00:07:58.326
With the Laplacian,

00:07:58.326 --> 00:07:59.508
you wanna start thinking about

00:07:59.508 --> 00:08:00.806
a circle of points around it,

00:08:00.806 --> 00:08:01.827
all of its neighbors,

00:08:01.827 --> 00:08:03.306
and in fact, think of a perfect circle,

00:08:03.306 --> 00:08:04.139
so all of the points

00:08:04.139 --> 00:08:05.984
that are a specified distance away.

00:08:05.984 --> 00:08:08.151
The question the Laplacian is asking is,

00:08:08.151 --> 00:08:10.582
"Hey, are those neighbor
points, on average,

00:08:10.582 --> 00:08:13.697
"greater than or less
than your original point?"

00:08:13.697 --> 00:08:14.530
And this is actually,

00:08:14.530 --> 00:08:16.081
this is how I introduced the Laplacian

00:08:16.081 --> 00:08:17.886
in the original video where I was giving

00:08:17.886 --> 00:08:20.580
kind of the intuition for the Laplacian,

00:08:20.580 --> 00:08:23.536
you're asking, "Do the
points around a given input

00:08:23.536 --> 00:08:26.061
"happen to be greater
than it or less than it?"

00:08:26.061 --> 00:08:27.170
And if you're looking at a point

00:08:27.170 --> 00:08:29.902
where the Laplacian of your function

00:08:29.902 --> 00:08:32.930
happens to be greater
than zero at some point,

00:08:32.930 --> 00:08:34.623
that would mean all of the neighbors

00:08:34.623 --> 00:08:37.298
tend to be, on average,
greater than your point.

00:08:37.298 --> 00:08:38.569
Whereas if you're looking at a point

00:08:38.569 --> 00:08:40.108
where the Laplacian of your function

00:08:40.108 --> 00:08:41.435
is less than zero,

00:08:41.435 --> 00:08:43.006
then all of those neighbors, on average,

00:08:43.006 --> 00:08:44.343
would be less than your point.

00:08:44.343 --> 00:08:45.431
So in particular, you know,

00:08:45.431 --> 00:08:47.263
if the Laplacian was less than zero,

00:08:47.263 --> 00:08:50.125
your point might look
like a local maximum.

00:08:50.125 --> 00:08:51.581
Or if the Laplacian was greater than zero,

00:08:51.581 --> 00:08:53.015
it might look like a local minimum,

00:08:53.015 --> 00:08:53.848
'cause all of its neighbors

00:08:53.848 --> 00:08:55.531
would be greater than where it is.

00:08:55.531 --> 00:08:56.554
But for harmonic functions,

00:08:56.554 --> 00:08:58.247
what makes them so special

00:08:58.247 --> 00:08:59.716
is that you're saying

00:08:59.716 --> 00:09:01.832
the value of the function itself,

00:09:01.832 --> 00:09:03.816
or the value of the
Laplacian of the function

00:09:03.816 --> 00:09:05.286
at every possible point

00:09:05.286 --> 00:09:06.679
is equal to zero.

00:09:06.679 --> 00:09:08.173
So no matter what point you choose,

00:09:08.173 --> 00:09:09.451
those neighbors are gonna be,

00:09:09.451 --> 00:09:12.070
on average, the same value as this guy.

00:09:12.070 --> 00:09:14.074
So the height of the graph
above those neighbors

00:09:14.074 --> 00:09:15.615
will, on average, be the same.

00:09:15.615 --> 00:09:17.204
So,

00:09:17.204 --> 00:09:18.786
so if we kind of look around the graph,

00:09:18.786 --> 00:09:20.315
what that should mean is,

00:09:20.315 --> 00:09:22.058
let's say you're looking
at an input point,

00:09:22.058 --> 00:09:24.144
you know, the output of this guy.

00:09:24.144 --> 00:09:25.461
And if we looked at all of its,

00:09:25.461 --> 00:09:26.460
the circle of its neighbors,

00:09:26.460 --> 00:09:28.711
and kind of projected
them up onto the graph,

00:09:28.711 --> 00:09:29.962
what it should mean is that the height

00:09:29.962 --> 00:09:31.435
of all the points on this circle,

00:09:31.435 --> 00:09:32.954
on average, are the same as that.

00:09:32.954 --> 00:09:34.152
And no matter where you look,

00:09:34.152 --> 00:09:35.877
that should kind of average out.

00:09:35.877 --> 00:09:37.493
And again, I encourage you to

00:09:37.493 --> 00:09:38.393
take a look at,

00:09:38.393 --> 00:09:39.309
take a look at this function

00:09:39.309 --> 00:09:40.606
and actually evaluate the Laplacian

00:09:40.606 --> 00:09:41.712
to see that it's zero.

00:09:41.712 --> 00:09:44.173
But what's interesting
is it's not at all clear,

00:09:44.173 --> 00:09:45.190
just looking at this E to the X

00:09:45.190 --> 00:09:46.898
times sine of Y formula,

00:09:46.898 --> 00:09:49.895
that the average value of
a circle of input points

00:09:49.895 --> 00:09:51.690
is always gonna kind of equal the value

00:09:51.690 --> 00:09:53.006
of the point at the center.

00:09:53.006 --> 00:09:54.485
That's not something you can easily tell

00:09:54.485 --> 00:09:56.040
just looking at that formula.

00:09:56.040 --> 00:09:58.217
But with what's not
that hard a computation,

00:09:58.217 --> 00:09:59.533
you can make this conclusion,

00:09:59.533 --> 00:10:00.803
which is pretty far-reaching.

00:10:00.803 --> 00:10:02.816
And this comes up all the time in physics.

00:10:02.816 --> 00:10:04.594
You know, for example, heat is one

00:10:04.594 --> 00:10:05.987
where maybe you wanna describe

00:10:05.987 --> 00:10:08.765
how the heat at a certain point in a room

00:10:08.765 --> 00:10:10.927
is related to the
average value of the heat

00:10:10.927 --> 00:10:12.806
of all of the points kind of around it.

00:10:12.806 --> 00:10:14.856
And in fact, it comes up in
all sorts of circumstances

00:10:14.856 --> 00:10:17.032
where you have some
point in physical space

00:10:17.032 --> 00:10:18.336
and something about that point,

00:10:18.336 --> 00:10:19.599
maybe like the rate at which

00:10:19.599 --> 00:10:21.276
some property of it is changing

00:10:21.276 --> 00:10:23.334
corresponds to the average value

00:10:23.334 --> 00:10:24.643
at points around it.

00:10:24.643 --> 00:10:26.577
So whenever you're sort
of relating neighbors

00:10:26.577 --> 00:10:28.787
to your original point,
the Laplacian comes in,

00:10:28.787 --> 00:10:30.558
and harmonic functions have this tendency

00:10:30.558 --> 00:10:32.670
to correspond to some notion of stability.

00:10:32.670 --> 00:10:34.244
And I won't go deeper into that now.

00:10:34.244 --> 00:10:35.531
This is really,

00:10:35.531 --> 00:10:36.742
that really starts to get into the topic

00:10:36.742 --> 00:10:38.651
of partial differential equations.

00:10:38.651 --> 00:10:39.921
But at least in the context of

00:10:39.921 --> 00:10:41.324
just multivariable calculus,

00:10:41.324 --> 00:10:42.660
I wanted to shed some light

00:10:42.660 --> 00:10:44.836
on interpreting this operator,

00:10:44.836 --> 00:10:45.669
and kind of interpreting

00:10:45.669 --> 00:10:48.340
the physical and geometric properties

00:10:48.340 --> 00:10:51.034
that that implies about a function.

00:10:51.034 --> 00:10:52.124
And with that,

00:10:52.124 --> 00:10:54.291
I will see you next video.


Le Chatelier's principle: Worked example

WEBVTT
Kind: captions
Language: en

00:00:01.307 --> 00:00:03.356
- [Voiceover] In this
video we're going to go

00:00:03.356 --> 00:00:05.356
through an example reaction

00:00:05.356 --> 00:00:07.893
that uses Le Chatelier's principle.

00:00:07.893 --> 00:00:09.457
So what we're gonna do is,

00:00:09.457 --> 00:00:11.788
we're gonna apply Le Chatelier's principle

00:00:11.788 --> 00:00:14.444
to look at various
changes to this reaction

00:00:14.444 --> 00:00:18.365
when we perturb our
reaction from equilibrium.

00:00:18.365 --> 00:00:20.029
Just as a reminder,

00:00:20.029 --> 00:00:22.470
what do I mean when I say,

00:00:22.470 --> 00:00:25.484
This reaction is at equilibrium.?

00:00:25.484 --> 00:00:28.715
So what that means is we
have a reversible reaction.

00:00:28.715 --> 00:00:31.978
We have the forward reaction
which has some rate K forward.

00:00:31.978 --> 00:00:35.728
The reverse reaction
has some rate K backward

00:00:36.875 --> 00:00:39.042
and a dynamic equilibrium.

00:00:40.187 --> 00:00:42.724
These rate are equal to each other,

00:00:42.724 --> 00:00:45.828
so all of the concentrations
are going to stay constant,

00:00:45.828 --> 00:00:49.672
and then what we do is we
decide to see what happens

00:00:49.672 --> 00:00:52.672
when we add some carbon dioxide gas.

00:00:54.049 --> 00:00:56.005
If we add carbon dioxide gas,

00:00:56.005 --> 00:00:58.858
the concentration of carbon
dioxide gas will go up,

00:00:58.858 --> 00:01:01.483
or you can think about it
as a partial pressure going up.

00:01:01.483 --> 00:01:03.389
And Le Chatelier's principle tells us,

00:01:03.389 --> 00:01:05.457
that if we had a reaction at equilibrium

00:01:05.457 --> 00:01:07.733
and then we perturbed
it by adding more CO2,

00:01:07.733 --> 00:01:11.900
it will shift to try to reduce
the effect of that change.

00:01:13.360 --> 00:01:17.527
It will favor the reverse
reaction, so if we add CO2,

00:01:18.437 --> 00:01:21.770
what happens is, we favor our reactants.

00:01:25.933 --> 00:01:27.766
Another way we can see this

00:01:27.766 --> 00:01:31.841
is by looking at the equilibrium
constant for this reaction.

00:01:31.841 --> 00:01:34.553
We can write our equilibrium constant K,

00:01:34.553 --> 00:01:36.387
where this is a capital K.

00:01:36.387 --> 00:01:38.053
It's kind of confusing,

00:01:38.053 --> 00:01:40.953
but I will try to make
this look like a capital K.

00:01:40.953 --> 00:01:43.036
(laughs)

00:01:43.930 --> 00:01:45.502
We can write it in two ways,

00:01:45.502 --> 00:01:49.628
we can write it in terms
of the molar concentration,

00:01:49.628 --> 00:01:51.456
and if we write Kc,

00:01:51.456 --> 00:01:54.613
the expression will be
the product concentration,

00:01:54.613 --> 00:01:57.094
our CO2 gas concentration.

00:01:57.094 --> 00:02:00.494
And that's it, because
when we write out Kc,

00:02:00.494 --> 00:02:02.788
we write out concentrations of gases

00:02:02.788 --> 00:02:05.633
and we write out
concentrations of solutions,

00:02:05.633 --> 00:02:07.672
but we don't include solids.

00:02:07.672 --> 00:02:11.395
Kc is just the concentration
of CO2 at equilibrium.

00:02:11.395 --> 00:02:13.123
I'm going to write an eq there

00:02:13.123 --> 00:02:15.294
just to show that's the
equilibrium concentration.

00:02:15.294 --> 00:02:17.091
I said that you can also write it

00:02:17.091 --> 00:02:20.148
in terms of partial pressures,

00:02:20.148 --> 00:02:24.979
so there's our fancy capital
K with a p subscript,

00:02:24.979 --> 00:02:27.487
which means that instead
of concentrations,

00:02:27.487 --> 00:02:29.375
we're writing everything for gases

00:02:29.375 --> 00:02:31.294
in terms of partial pressures.

00:02:31.294 --> 00:02:34.298
We have the partial
pressure of CO2 and again,

00:02:34.298 --> 00:02:36.747
that's it, because
everything else is a solid,

00:02:36.747 --> 00:02:40.096
so we don't include those in
our equilibrium expression.

00:02:40.096 --> 00:02:42.910
Writing these expressions
out will be really helpful

00:02:42.910 --> 00:02:44.667
for our second condition.

00:02:44.667 --> 00:02:47.434
We're going to think about what happens,

00:02:47.434 --> 00:02:49.113
when you increase

00:02:49.113 --> 00:02:52.190
the volume of our container.

00:02:52.190 --> 00:02:54.720
We can rewrite the partial pressure,

00:02:54.720 --> 00:02:57.215
actually, in terms of the volume,

00:02:57.215 --> 00:03:00.289
so if you use the ideal gas law,

00:03:00.289 --> 00:03:03.289
the partial pressure of CO2 is equal to

00:03:03.289 --> 00:03:04.805
the moles of CO2

00:03:04.805 --> 00:03:06.938
times RT

00:03:06.938 --> 00:03:08.771
divided by the volume.

00:03:09.992 --> 00:03:12.800
Similarly, we can rewrite
molar concentration in terms

00:03:12.800 --> 00:03:14.386
of moles divided by volume.

00:03:14.386 --> 00:03:16.836
If we increase the
volume of our container,

00:03:16.836 --> 00:03:19.735
increasing the volume, since
it's in the denominator,

00:03:19.735 --> 00:03:21.776
will make our pressure go down,

00:03:21.776 --> 00:03:24.306
so our partial pressure
of CO2 will go down

00:03:24.306 --> 00:03:26.592
and we won't be at equilibrium anymore

00:03:26.592 --> 00:03:28.716
and the same for the concentration,

00:03:28.716 --> 00:03:30.104
since our perturbation

00:03:30.104 --> 00:03:33.005
is decreasing the carbon
dioxide concentrations,

00:03:33.005 --> 00:03:34.883
Le Chatelier's principle says,

00:03:34.883 --> 00:03:38.722
that our reaction will try
to counteract that change.

00:03:38.722 --> 00:03:40.965
It will try to get back to equilibrium

00:03:40.965 --> 00:03:43.616
and try to get the CO2
concentration back up,

00:03:43.616 --> 00:03:46.449
so it will have to favor products.

00:03:48.968 --> 00:03:52.441
Our reaction will favor
the products to try to get

00:03:52.441 --> 00:03:54.073
the moles of CO2 back up,

00:03:54.073 --> 00:03:58.581
so that we get back to the
equilibrium concentration of CO2

00:03:58.581 --> 00:04:00.508
and the equilibrium partial pressure.

00:04:00.508 --> 00:04:01.847
The third change we're gonna look at

00:04:01.847 --> 00:04:05.097
is what happens when you add argon gas.

00:04:07.644 --> 00:04:09.811
Argon gas is an inert gas.

00:04:10.658 --> 00:04:13.778
We don't expect it to react to anything.

00:04:13.778 --> 00:04:18.280
One thing that will happen
when you add the argon gas is

00:04:18.280 --> 00:04:22.447
it will increase the overall
pressure of your container,

00:04:23.524 --> 00:04:26.774
so it will increase the total pressure.

00:04:28.894 --> 00:04:30.842
But that actually doesn't tell us

00:04:30.842 --> 00:04:32.885
what it does to our equilibrium.

00:04:32.885 --> 00:04:37.341
Let's look back at our
equilibrium expressions Kc and Kp.

00:04:37.341 --> 00:04:41.382
We can see that the partial
pressure for Kp only depends

00:04:41.382 --> 00:04:44.653
on the moles of our CO2 and our volume.

00:04:44.653 --> 00:04:47.306
Since we didn't change the moles of CO2

00:04:47.306 --> 00:04:49.506
and we also didn't change the volume.

00:04:49.506 --> 00:04:51.424
Even though we increased P-total,

00:04:51.424 --> 00:04:54.976
the partial pressure
of CO2 stayed the same.

00:04:54.976 --> 00:04:59.143
That means we didn't perturb
our reaction from equilibrium

00:04:59.143 --> 00:05:02.903
and since we didn't perturb
it from equilibrium,

00:05:02.903 --> 00:05:04.948
there'll be no shift.

00:05:04.948 --> 00:05:07.276
We are still at equilibrium.

00:05:07.276 --> 00:05:10.338
The concentrations will
still stay the same.

00:05:10.338 --> 00:05:14.338
What happens when we add
more calcium carbonate?

00:05:16.214 --> 00:05:19.482
That's our starting
material and it is a solid.

00:05:19.482 --> 00:05:21.733
Our equilibrium expressions are determined

00:05:21.733 --> 00:05:23.261
by our CO2 concentrations,

00:05:23.261 --> 00:05:24.973
so adding more calcium carbonate,

00:05:24.973 --> 00:05:26.197
which is a solid,

00:05:26.197 --> 00:05:28.366
isn't actually going
to perturb our reaction

00:05:28.366 --> 00:05:29.346
from equilibrium.

00:05:29.346 --> 00:05:31.636
Our reaction is still
going to be at equilibrium

00:05:31.636 --> 00:05:35.219
and we will get no
shift in concentrations.

00:05:36.290 --> 00:05:39.155
We're actually gonna
look at one more thing.

00:05:39.155 --> 00:05:41.483
We're going to think about what happens

00:05:41.483 --> 00:05:43.483
when you add a catalyst.

00:05:45.688 --> 00:05:48.306
Let's say we want to
speed up this reaction.

00:05:48.306 --> 00:05:50.515
We can envision what's going on here

00:05:50.515 --> 00:05:54.682
when we add the catalyst
by using an energy diagram.

00:05:56.685 --> 00:05:59.093
If we have an energy diagram.

00:05:59.093 --> 00:06:01.744
We have energy on the Y axis

00:06:01.744 --> 00:06:04.605
and we're looking at
the difference in energy

00:06:04.605 --> 00:06:08.936
between our reactants,
or starting materials,

00:06:08.936 --> 00:06:10.353
with our products

00:06:12.370 --> 00:06:15.432
and I just sort of made up
these relative energies.

00:06:15.432 --> 00:06:17.269
The way that I have this drawn here,

00:06:17.269 --> 00:06:19.024
we can see that our product

00:06:19.024 --> 00:06:21.760
is lower energy than our starting material

00:06:21.760 --> 00:06:25.967
and our forward rate,
Kf, which is is up here,

00:06:25.967 --> 00:06:29.296
is determined by the size
of this activation barrier

00:06:29.296 --> 00:06:32.078
between our starting material
and our transition state.

00:06:32.078 --> 00:06:33.911
Our backward rate, Kb,

00:06:36.313 --> 00:06:38.120
is determined by the size

00:06:38.120 --> 00:06:40.491
of this energy barrier,

00:06:40.491 --> 00:06:42.125
so the difference in energy

00:06:42.125 --> 00:06:44.819
between the product and
our transition state.

00:06:44.819 --> 00:06:47.191
If we add a catalyst to our reaction,

00:06:47.191 --> 00:06:51.234
we can think about it as
lowering the activation energy

00:06:51.234 --> 00:06:52.567
for our reaction

00:06:54.505 --> 00:06:57.812
and that means we have
a lower energy barrier

00:06:57.812 --> 00:06:59.448
for our forward reaction,

00:06:59.448 --> 00:07:02.714
so our forward reaction
is gonna get faster,

00:07:02.714 --> 00:07:06.468
but it's also gonna lower the
rate of our backward reaction.

00:07:06.468 --> 00:07:09.801
K backward, or Kb, is also gonna speedup

00:07:11.113 --> 00:07:13.228
and since it speeds up both the forward

00:07:13.228 --> 00:07:14.862
and the backward reactions

00:07:14.862 --> 00:07:17.723
adding a catalyst also
won't perturb your reaction

00:07:17.723 --> 00:07:19.140
from equilibrium.

00:07:20.094 --> 00:07:24.261
Adding a catalyst will result
in no shift in concentrations.

00:07:27.299 --> 00:07:30.441
So the main things to remember
from this problem, I think.

00:07:30.441 --> 00:07:32.895
the things I find most tricky anyway,

00:07:32.895 --> 00:07:35.387
are that adding an inert gas,

00:07:35.387 --> 00:07:37.467
it'll increase your total pressure,

00:07:37.467 --> 00:07:40.204
but it won't actually change
any of your partial pressures.

00:07:40.204 --> 00:07:42.980
It won't shift your
reaction from equilibrium

00:07:42.980 --> 00:07:47.147
and the same thing is true
for solids and catalysts.

00:07:48.099 --> 00:07:50.671
All of those three things: inert gases,

00:07:50.671 --> 00:07:53.286
solids, and catalysts,

00:07:53.286 --> 00:07:56.771
will not shift your
reaction from equilibrium.


Cutting shapes into equal parts

WEBVTT
Kind: captions
Language: en

00:00:01.491 --> 00:00:03.737
- [Voiceover] Is each piece equal to

00:00:03.737 --> 00:00:06.904
1/4 of the area of the pie?

00:00:08.187 --> 00:00:09.375
So we have a pie,

00:00:09.375 --> 00:00:13.378
and it has one, two, three, four pieces.

00:00:13.378 --> 00:00:14.896
So it does have four pieces.

00:00:14.896 --> 00:00:19.424
So is one of those pieces
equal to 1/4 of the pie?

00:00:19.424 --> 00:00:20.937
Well let's talk about what we mean

00:00:20.937 --> 00:00:24.289
when we have a fraction like 1/4.

00:00:24.289 --> 00:00:26.946
The one in the fraction, the numerator,

00:00:26.946 --> 00:00:28.538
represents a number of pieces.

00:00:28.538 --> 00:00:30.627
So here, one piece.

00:00:30.627 --> 00:00:32.752
One piece of pie.

00:00:32.752 --> 00:00:35.344
And then the four, when
we're talking about fractions

00:00:35.344 --> 00:00:38.749
is always talking about
the number of equal size.

00:00:38.749 --> 00:00:39.582
Equal

00:00:41.641 --> 00:00:42.474
size

00:00:43.686 --> 00:00:44.519
pieces.

00:00:45.697 --> 00:00:48.400
So in this case four equal size pieces.

00:00:48.400 --> 00:00:50.816
So the question is,

00:00:50.816 --> 00:00:54.945
is each piece one of
four equal size pieces?

00:00:54.945 --> 00:00:56.699
Let's look at the pie.

00:00:56.699 --> 00:00:58.539
I think it's pretty clear

00:00:58.539 --> 00:01:00.420
that these pieces on the end

00:01:00.420 --> 00:01:03.452
are not equal, they are smaller

00:01:03.452 --> 00:01:05.333
than the two pieces in the middle.

00:01:05.333 --> 00:01:07.302
If you love cherry pie,

00:01:07.302 --> 00:01:09.918
you are not happy about
getting this end piece.

00:01:09.918 --> 00:01:11.636
Because it is smaller.

00:01:11.636 --> 00:01:14.960
It is not an equal size piece.

00:01:14.960 --> 00:01:19.359
So yes, each piece is
one out of four pieces.

00:01:19.359 --> 00:01:23.807
But it is not one of
four equal size pieces.

00:01:23.807 --> 00:01:26.429
Therefore it is not 1/4.

00:01:26.429 --> 00:01:28.922
So our answer is no.

00:01:28.922 --> 00:01:30.595
No, no, no.

00:01:30.595 --> 00:01:32.665
Each piece is not 1/4

00:01:32.665 --> 00:01:35.082
or an equal share of the pie.


Mesh current method (step 4 solve)

WEBVTT
Kind: captions
Language: en

00:00:01.109 --> 00:00:03.405
- [Voiceover] We're working
on the mesh current method,

00:00:03.405 --> 00:00:05.728
a method of analyzing circuits.

00:00:05.728 --> 00:00:08.773
And in the previous video
we set up our circuit,

00:00:08.773 --> 00:00:12.940
we set up our mesh currents
flowing around these loops

00:00:14.079 --> 00:00:18.024
within the circuit, and we
solved for the easy currents.

00:00:18.024 --> 00:00:20.589
That was the current on this side,

00:00:20.589 --> 00:00:24.257
this current source constrains
i3 to be a fixed value,

00:00:24.257 --> 00:00:25.859
so we already know i3.

00:00:25.859 --> 00:00:28.110
And then we created two equations

00:00:28.110 --> 00:00:32.767
from the two independent
variables, the two currents here,

00:00:32.767 --> 00:00:34.699
to describe how the circuit works.

00:00:34.699 --> 00:00:38.866
And in this video we're going
to solve these two equations.

00:00:40.016 --> 00:00:41.572
One thing to remind ourselves,

00:00:41.572 --> 00:00:43.731
when we defined these mesh currents

00:00:43.731 --> 00:00:45.984
all of them went in the same direction.

00:00:45.984 --> 00:00:48.188
It helps to make them all
go in the same direction,

00:00:48.188 --> 00:00:50.255
but all clockwise in this case.

00:00:50.255 --> 00:00:51.696
And that's so that when you have

00:00:51.696 --> 00:00:53.111
this little pattern down here

00:00:53.111 --> 00:00:55.694
of i1 minus i2, or i2 minus i3,

00:00:56.711 --> 00:00:58.591
that it becomes a familiar pattern,

00:00:58.591 --> 00:01:01.586
and it helps to get the signs right.

00:01:01.586 --> 00:01:03.026
The other thing to point out is

00:01:03.026 --> 00:01:06.093
the reason we're using
meshes and not any other loop

00:01:06.093 --> 00:01:08.181
in this circuit, not overlapping loops

00:01:08.181 --> 00:01:09.551
like we could have gone all the way

00:01:09.551 --> 00:01:11.557
around the outside in a loop.

00:01:11.557 --> 00:01:12.979
The reason that meshes,

00:01:12.979 --> 00:01:16.023
in particular the open
areas of the circuit,

00:01:16.023 --> 00:01:18.270
are a good way to do this method

00:01:18.270 --> 00:01:20.406
is because it produces exactly

00:01:20.406 --> 00:01:23.402
the right number of equations
to solve the circuit.

00:01:23.402 --> 00:01:24.888
You don't have too few equations,

00:01:24.888 --> 00:01:28.070
and you don't have too many equations.

00:01:28.070 --> 00:01:29.229
So let me move the screen up

00:01:29.229 --> 00:01:32.562
and we'll go to work on these equations.

00:01:36.150 --> 00:01:38.982
So now we begin step four,

00:01:38.982 --> 00:01:41.745
which is to solve the equations.

00:01:41.745 --> 00:01:42.860
And what I want to do first is,

00:01:42.860 --> 00:01:44.184
just to tidy up a little bit,

00:01:44.184 --> 00:01:45.903
I'm gonna take the constants

00:01:45.903 --> 00:01:47.875
and I'm gonna put them
on the right hand side,

00:01:47.875 --> 00:01:50.755
and I'm gonna gather
together the terms for i1

00:01:50.755 --> 00:01:53.958
and the terms for i2 in separate places.

00:01:53.958 --> 00:01:57.279
So if we go at the first
equation and we look for i1,

00:01:57.279 --> 00:01:59.832
it's mentioned here twice,

00:01:59.832 --> 00:02:01.082
so we get minus

00:02:03.315 --> 00:02:05.648
and it's a combination of r1

00:02:07.273 --> 00:02:09.023
and plus r2 times i1.

00:02:15.437 --> 00:02:19.020
And for the i2 term,
let's see if i2 shows up,

00:02:19.020 --> 00:02:21.522
i2 just shows up once here.

00:02:21.522 --> 00:02:23.855
So it's gonna be plus r2 i2.

00:02:29.533 --> 00:02:31.343
And that's gonna be equal to,

00:02:31.343 --> 00:02:33.665
let's move V over to the other side,

00:02:33.665 --> 00:02:36.248
so V becomes minus V over here.

00:02:37.334 --> 00:02:40.050
So that's our first equation.

00:02:40.050 --> 00:02:42.583
So now let's tidy up the second equation.

00:02:42.583 --> 00:02:46.250
The i1 terms, this is
the only i1 term here.

00:02:47.086 --> 00:02:49.169
So it's plus r2 i2, oops,

00:02:57.460 --> 00:03:00.460
plus r2 i1, that's the only i1 term.

00:03:03.933 --> 00:03:05.607
Now if we do i2 terms,

00:03:05.607 --> 00:03:09.774
we're gonna have minus r2 i2,
minus r3 i2, and minus r4, i2.

00:03:15.533 --> 00:03:17.700
So let's do that this way,

00:03:22.457 --> 00:03:23.290
r2 plus r3

00:03:28.731 --> 00:03:29.564
plus r4

00:03:34.198 --> 00:03:35.031
times i2.

00:03:38.781 --> 00:03:40.942
Now we have an i3 term in here,

00:03:40.942 --> 00:03:45.215
but we know what i3 is,
i3 we know from this.

00:03:45.215 --> 00:03:49.139
So I'm gonna write that down right here,

00:03:49.139 --> 00:03:53.387
that equals minus I, there's
lots of minus signs here,

00:03:53.387 --> 00:03:55.732
so let's see if we can get this right.

00:03:55.732 --> 00:03:59.232
Minus r4 times minus i3 is positive r4 i3,

00:04:02.906 --> 00:04:04.739
or it's negative r4 I,

00:04:07.692 --> 00:04:11.859
and when we take it to the
other side it'll be plus r4 I.

00:04:19.045 --> 00:04:20.345
Let me check that again.

00:04:20.345 --> 00:04:23.928
Minus r4 times minus
i3 is a positive sign,

00:04:26.336 --> 00:04:28.566
I gives it a negative
sign, and when we take it

00:04:28.566 --> 00:04:31.128
to the other side it gets a positive sign.

00:04:31.128 --> 00:04:32.685
Good.

00:04:32.685 --> 00:04:33.684
Alright?

00:04:33.684 --> 00:04:37.932
So this is what our
equations look like now.

00:04:37.932 --> 00:04:39.349
Let's keep going.

00:04:41.067 --> 00:04:43.528
I can continue using symbols
for the resistor values,

00:04:43.528 --> 00:04:45.572
but it's gonna get kind of complicated

00:04:45.572 --> 00:04:48.148
and it doesn't help
with our understanding,

00:04:48.148 --> 00:04:52.065
so I'm gonna switch over
to a specific circuit.

00:04:59.402 --> 00:05:02.119
So what I did was I filled
in some resistor values here

00:05:02.119 --> 00:05:04.278
and I put five volts
on the voltage source,

00:05:04.278 --> 00:05:07.028
one k ohms for r1, 2k ohms is r2,

00:05:07.878 --> 00:05:10.295
3k ohms is r3, 3k ohms is r4,

00:05:12.916 --> 00:05:15.563
and there's two milliamps over here.

00:05:15.563 --> 00:05:17.118
So I'm just gonna write these values

00:05:17.118 --> 00:05:20.277
onto the equation here so I don't mess up.

00:05:20.277 --> 00:05:23.527
One k, two k, two k, that's five volts.

00:05:27.432 --> 00:05:31.099
That's two k ohms, two
k ohms, three k ohms,

00:05:32.749 --> 00:05:35.416
and r4 was also three, is three,

00:05:39.127 --> 00:05:41.044
and I is two milliamps.

00:05:49.409 --> 00:05:53.576
I'm gonna move the circuit down
out of the way a little bit.

00:05:54.540 --> 00:05:56.212
Now we have room for our equation.

00:05:56.212 --> 00:05:58.975
Let's fill it out again
with the real values.

00:05:58.975 --> 00:06:02.975
So it's gonna be minus
one plus two is three i1,

00:06:06.058 --> 00:06:06.975
plus two i2

00:06:12.541 --> 00:06:14.041
equals minus five.

00:06:18.624 --> 00:06:20.807
We know all the resistors are k ohms

00:06:20.807 --> 00:06:23.662
and so we can actually
scale our equations down

00:06:23.662 --> 00:06:26.393
by a factor of 1,000 and just use

00:06:26.393 --> 00:06:28.704
the simple numbers without the ks.

00:06:28.704 --> 00:06:31.371
So our second equation is two i1

00:06:35.769 --> 00:06:39.436
minus two plus three
plus three is eight i2,

00:06:43.886 --> 00:06:47.886
and that equals three k
ohms times two milliamps,

00:06:47.886 --> 00:06:48.886
is plus six.

00:06:55.241 --> 00:06:59.408
So this is getting close
to something we can solve.

00:07:02.512 --> 00:07:04.321
So I think the way I want to solve this

00:07:04.321 --> 00:07:08.083
is I'm going to take this equation here

00:07:08.083 --> 00:07:10.083
and multiply it by four,

00:07:11.311 --> 00:07:14.478
and then add it to this equation here.

00:07:17.720 --> 00:07:20.065
And let's see what we get.

00:07:20.065 --> 00:07:23.232
So four times minus three is minus 12,

00:07:25.011 --> 00:07:27.844
and we add it and we're gonna get.

00:07:31.024 --> 00:07:35.191
Four times two is eight,
minus eight, and i2 drops out.

00:07:37.224 --> 00:07:39.547
That was the purpose of the four.

00:07:39.547 --> 00:07:43.714
And that equals four times
minus five is minus 20,

00:07:44.794 --> 00:07:46.544
plus six is minus 14.

00:07:50.599 --> 00:07:53.099
So i1 equals minus 14 over 10,

00:07:57.762 --> 00:08:00.012
or i1 equals 1.4 milliamps,

00:08:06.469 --> 00:08:08.185
and that's cool.

00:08:08.185 --> 00:08:10.852
We get to put a box around that.

00:08:13.515 --> 00:08:16.602
So as a reminder, we defined
i1 to be the mesh current

00:08:16.602 --> 00:08:19.185
flowing around like this in i1.

00:08:21.527 --> 00:08:22.694
1.4 milliamps.

00:08:25.687 --> 00:08:29.232
That's one of our independent variables.

00:08:29.232 --> 00:08:31.299
Now that we know i1, let's just choose one

00:08:31.299 --> 00:08:35.990
of our original equations here
and figure out what i2 is.

00:08:35.990 --> 00:08:40.157
We'll plug in two times
i1, which is 1.4 milliamps,

00:08:43.560 --> 00:08:45.727
minus eight i2 equals six.

00:08:50.083 --> 00:08:52.987
And now we have to figure out i2.

00:08:52.987 --> 00:08:55.030
So let's solve this.

00:08:55.030 --> 00:08:57.113
Minus eight i2 equals six

00:09:00.555 --> 00:09:02.805
minus two times 1.4 is 2.8.

00:09:06.824 --> 00:09:09.907
i2 equals 3.2 divided by minus eight,

00:09:17.157 --> 00:09:19.907
or i2 equals minus 0.4 milliamps.

00:09:27.777 --> 00:09:31.214
And we can draw a box around that too.

00:09:31.214 --> 00:09:33.095
So let's put that mesh current in.

00:09:33.095 --> 00:09:35.178
And our mesh current was.

00:09:38.886 --> 00:09:42.969
That was i2, and that
equals minus 0.4 milliamps.

00:09:49.289 --> 00:09:52.161
And what that means is
that the current is flowing

00:09:52.161 --> 00:09:53.915
in the negative direction,

00:09:53.915 --> 00:09:57.622
so it's actually flowing this
way through these resistors.

00:09:57.622 --> 00:10:00.501
It's actually flowing in the
counterclockwise direction

00:10:00.501 --> 00:10:02.311
is the positive current.

00:10:02.311 --> 00:10:05.228
And from before we already knew i3.

00:10:09.038 --> 00:10:11.538
i3 equals minus two milliamps.

00:10:16.142 --> 00:10:17.583
So let's do one last thing here,

00:10:17.583 --> 00:10:19.996
let's figure out the
currents in the elements.

00:10:19.996 --> 00:10:22.032
Let's just figure out a couple of them,

00:10:22.032 --> 00:10:23.865
that one and that one.

00:10:24.723 --> 00:10:27.640
So that is ir2 and this one is ir4.

00:10:32.875 --> 00:10:34.542
ir2 equals i2 minus,

00:10:42.591 --> 00:10:43.424
oops,

00:10:46.876 --> 00:10:48.459
i1 minus i2 equals,

00:10:52.984 --> 00:10:55.317
i1 is 1.4 minus negative .4,

00:11:04.325 --> 00:11:07.742
and so that current equals 1.8 milliamps.

00:11:14.040 --> 00:11:18.207
And we'll do one last one,
we'll do ir4, this is r4 here.

00:11:27.044 --> 00:11:28.961
And ir4 is i2 minus i3.

00:11:37.771 --> 00:11:41.188
So that equals, i2 is minus .4, minus i3.

00:11:44.460 --> 00:11:46.627
i3 is minus two milliamps.

00:11:50.705 --> 00:11:51.788
So ir4 equals

00:11:58.216 --> 00:12:00.799
two minus .4, or 1.6 milliamps.

00:12:07.829 --> 00:12:08.828
And there you have it,

00:12:08.828 --> 00:12:11.796
we used the mesh current
method to solve our circuit.


Mesh current steps 1 to 3

WEBVTT
Kind: captions
Language: en

00:00:02.153 --> 00:00:03.639
- [Voiceover] Now we're
gonna discuss the second

00:00:03.639 --> 00:00:06.730
of two popular ways to analyze circuits,

00:00:06.730 --> 00:00:09.614
and this one is called
the mesh current method.

00:00:09.614 --> 00:00:11.526
This is actually one of my favorite.

00:00:11.526 --> 00:00:13.828
There's a fun spot here
where we make up currents

00:00:13.828 --> 00:00:17.035
flowing around in circles
inside the circuit,

00:00:17.035 --> 00:00:18.785
and it's interesting.

00:00:19.819 --> 00:00:23.078
In the mesh current method,
what we do is we define

00:00:23.078 --> 00:00:26.579
mesh currents, so we
need to define that word.

00:00:26.579 --> 00:00:27.614
A mesh current.

00:00:27.614 --> 00:00:30.814
A mesh current is a current
that flows in a mesh.

00:00:30.814 --> 00:00:34.349
And a mesh is the open
windows of the circuit.

00:00:34.349 --> 00:00:36.546
Here's one of the meshes in our circuit.

00:00:36.546 --> 00:00:39.673
There's another mesh
here, it's an open window,

00:00:39.673 --> 00:00:43.502
and this third one here is another one.

00:00:43.502 --> 00:00:45.588
And this will be our
mesh current number one.

00:00:45.588 --> 00:00:47.816
We'll call it i1, and we can define these

00:00:47.816 --> 00:00:50.399
other mesh currents, like this.

00:00:52.855 --> 00:00:54.891
And I'm gonna define them all to go around

00:00:54.891 --> 00:00:55.858
the same direction.

00:00:55.858 --> 00:00:57.617
They're all going clockwise,

00:00:57.617 --> 00:00:59.200
around our circuit.

00:01:03.644 --> 00:01:05.248
Now let me take a second
to label the other

00:01:05.248 --> 00:01:07.045
components in our circuit here.

00:01:07.045 --> 00:01:10.212
We'll call this V, we'll call this R1,

00:01:11.601 --> 00:01:12.434
R2,

00:01:14.321 --> 00:01:15.154
R3,

00:01:16.875 --> 00:01:18.339
R4,

00:01:18.339 --> 00:01:20.114
and this has a current source in it.

00:01:20.114 --> 00:01:22.225
We'll call that I.

00:01:22.225 --> 00:01:23.872
Makes it a little bit more interesting,

00:01:23.872 --> 00:01:25.436
and this is one of the reasons we use

00:01:25.436 --> 00:01:27.642
the mesh current method.

00:01:27.642 --> 00:01:30.713
The word mesh actually
comes from like screen doors

00:01:30.713 --> 00:01:33.753
or window screens where
there's wires and there's

00:01:33.753 --> 00:01:37.868
spaces between the wires,
so a space in a wire network

00:01:37.868 --> 00:01:39.096
is called a mesh.

00:01:39.096 --> 00:01:41.712
That's where the name comes from.

00:01:41.712 --> 00:01:45.017
These mesh currents are
kind of interesting.

00:01:45.017 --> 00:01:46.768
In some sense they exist, and then,

00:01:46.768 --> 00:01:48.901
but it doesn't seem like they really do.

00:01:48.901 --> 00:01:51.715
How can these currents be
flowing around in circles?

00:01:51.715 --> 00:01:53.407
Let's look a little closely.

00:01:53.407 --> 00:01:55.816
It's pretty clear that the
current flowing through

00:01:55.816 --> 00:01:59.983
our voltage source, V, is
equal to mesh current i1.

00:02:00.884 --> 00:02:04.267
And also, when we get up
here to R1, mesh current i1

00:02:04.267 --> 00:02:08.017
is flowing through R1
in this direction here.

00:02:09.254 --> 00:02:10.610
What happens when we get to R2?

00:02:10.610 --> 00:02:13.924
We have one mesh current
flowing down through R2,

00:02:13.924 --> 00:02:16.563
and another mesh current
flowing up through R2.

00:02:16.563 --> 00:02:18.757
How can this make sense?

00:02:18.757 --> 00:02:22.674
Let me go over here, and
we'll draw, redraw R2.

00:02:24.391 --> 00:02:28.317
Here's R2, and now I'll
redraw the mesh currents.

00:02:28.317 --> 00:02:32.433
So we have a mesh current,
i1 going down like that,

00:02:32.433 --> 00:02:34.909
and we have a mesh current,

00:02:34.909 --> 00:02:36.076
i2, like that.

00:02:37.624 --> 00:02:41.538
And I'll label on here, I'll
label the element current,

00:02:41.538 --> 00:02:43.871
for R2, we'll call that iR2.

00:02:47.697 --> 00:02:49.721
And what's gonna happen,
this is a really wonderful

00:02:49.721 --> 00:02:52.955
application of the
principle of superposition.

00:02:52.955 --> 00:02:56.872
i1 and i2 superimposed
inside R2 to create iR2.

00:03:01.066 --> 00:03:03.483
So if we wanted to know iR2,

00:03:05.803 --> 00:03:07.614
it would be the sum of these two currents.

00:03:07.614 --> 00:03:12.167
Now i1 is flowing in the
same direction as iR2,

00:03:12.167 --> 00:03:16.097
so this equals i1, and i2
is going in the opposite

00:03:16.097 --> 00:03:19.014
direction, so it gets a minus sign.

00:03:20.917 --> 00:03:23.392
And if we somehow knew
i1 and i2, this is how

00:03:23.392 --> 00:03:25.707
we would compute the actually current

00:03:25.707 --> 00:03:28.147
that's flowing in R2.

00:03:28.147 --> 00:03:30.159
Next, since this is a
method, I want to list out

00:03:30.159 --> 00:03:32.159
the steps of the method.

00:03:33.364 --> 00:03:35.700
The mesh current method is the first thing

00:03:35.700 --> 00:03:37.533
is to draw the meshes.

00:03:43.632 --> 00:03:47.799
And we did that over here,
when we drew i1, i2, and i3.

00:03:49.111 --> 00:03:52.944
And the second step is
to solve the easy ones.

00:03:59.310 --> 00:04:02.349
When we say an easy one, here's
an example of an easy one.

00:04:02.349 --> 00:04:06.516
The current source I, right
here, defines what i3 is.

00:04:07.656 --> 00:04:12.014
So i3 is an example of an
easy current to solve for.

00:04:12.014 --> 00:04:16.157
And then the third step in the method is

00:04:16.157 --> 00:04:16.990
to write

00:04:19.552 --> 00:04:21.434
Kirchhoff's voltage law.

00:04:21.434 --> 00:04:22.267
KVL.

00:04:23.492 --> 00:04:25.379
Write Kirchoff's voltage law equations,

00:04:25.379 --> 00:04:28.840
for the two remaining mesh currents.

00:04:28.840 --> 00:04:31.257
And the fourth step is solve.

00:04:34.597 --> 00:04:36.859
So in this video we're gonna
do the first three steps,

00:04:36.859 --> 00:04:40.681
and then in the next video,
we'll go the full solve.

00:04:40.681 --> 00:04:43.661
Okay, so let's run our
mesh current method.

00:04:43.661 --> 00:04:45.363
Number one, draw the meshes.

00:04:45.363 --> 00:04:48.521
Here's step one, happening right here.

00:04:48.521 --> 00:04:50.641
So we could check off, we did step one.

00:04:50.641 --> 00:04:52.462
Step two is solve for the easy ones.

00:04:52.462 --> 00:04:55.922
And with think, i3 counts as an easy one.

00:04:55.922 --> 00:04:59.422
i3 flows down, through the current source,

00:05:00.390 --> 00:05:02.390
and so, step number two,

00:05:05.213 --> 00:05:06.213
so step two,

00:05:08.069 --> 00:05:10.069
is i3 equals negative I,

00:05:14.275 --> 00:05:16.919
because they're pointing
in opposite directions.

00:05:16.919 --> 00:05:19.029
But that's it, that's the easy one.

00:05:19.029 --> 00:05:20.316
We've done that.

00:05:20.316 --> 00:05:22.649
Now we get to the next step.

00:05:25.801 --> 00:05:27.332
So now let's start step three.

00:05:27.332 --> 00:05:31.733
We're gonna write KVL equations
on each of these meshes.

00:05:31.733 --> 00:05:34.733
We'll start here, and we're
gonna go around this mesh

00:05:34.733 --> 00:05:37.804
in this direction and write
Kirchhoff's voltage law.

00:05:37.804 --> 00:05:40.727
So the first thing we
come to is a voltage rise.

00:05:40.727 --> 00:05:43.516
We go in the minus side and out the plus.

00:05:43.516 --> 00:05:46.266
That's a voltage rise of V volts.

00:05:49.028 --> 00:05:51.918
Next we get to R1, and R1 has
a current flowing through it,

00:05:51.918 --> 00:05:56.241
a mesh current, and it will
have a voltage on it like that,

00:05:56.241 --> 00:05:59.384
because the current's
coming in from this side.

00:05:59.384 --> 00:06:02.067
And that will give us a
voltage, we go in the plus,

00:06:02.067 --> 00:06:03.227
out the minus, so that's
gonna give us a voltage drop

00:06:03.227 --> 00:06:04.060
of...

00:06:06.914 --> 00:06:08.747
a drop of R1 times i1.

00:06:14.914 --> 00:06:15.837
Right.

00:06:15.837 --> 00:06:17.937
Now we're over to here,
we go around the corner,

00:06:17.937 --> 00:06:19.779
and we go through R2.

00:06:19.779 --> 00:06:23.029
Now, R2 is odd because it has i1 and i2

00:06:24.082 --> 00:06:26.322
flowing into it, so what are we gonna do?

00:06:26.322 --> 00:06:30.781
Well we already figured
out this element current

00:06:30.781 --> 00:06:33.870
is written down right over here.

00:06:33.870 --> 00:06:35.652
There it is right there.

00:06:35.652 --> 00:06:39.283
So if I label the current going down here,

00:06:39.283 --> 00:06:41.212
I have a plus and a minus,

00:06:41.212 --> 00:06:43.419
this is gonna be another voltage drop

00:06:43.419 --> 00:06:45.987
going around the loop.

00:06:45.987 --> 00:06:48.654
So it's gonna have a value of R2

00:06:51.045 --> 00:06:52.878
times this value here.

00:06:54.595 --> 00:06:55.595
i1 minus i2.

00:07:01.104 --> 00:07:03.239
And now we've made it
all the way back to home,

00:07:03.239 --> 00:07:04.594
to where we started from,

00:07:04.594 --> 00:07:06.704
and so that's our KVL equation,

00:07:06.704 --> 00:07:08.954
which we set equal to zero.

00:07:10.544 --> 00:07:11.966
All right.

00:07:11.966 --> 00:07:14.585
Next step, let's do the other mesh.

00:07:14.585 --> 00:07:16.119
We're gonna go start
here, and we're gonna go

00:07:16.119 --> 00:07:18.952
around this mesh, in the same way.

00:07:20.454 --> 00:07:23.713
First, the first term,
we're gonna see a voltage,

00:07:23.713 --> 00:07:26.130
oh look, it's a voltage rise,

00:07:27.247 --> 00:07:29.229
so I'll have a plus here.

00:07:29.229 --> 00:07:31.589
A voltage rise, and it's the same term

00:07:31.589 --> 00:07:33.364
that we had in the previous equation.

00:07:33.364 --> 00:07:34.197
It'll be plus R2

00:07:34.197 --> 00:07:35.697
times i1 minus i2.

00:07:42.428 --> 00:07:46.155
That's the voltage rise
going from here to here.

00:07:46.155 --> 00:07:47.441
Now we go through R3.

00:07:47.441 --> 00:07:49.210
Now R3 just has one mesh current.

00:07:49.210 --> 00:07:51.463
It just has i2 flowing through it.

00:07:51.463 --> 00:07:53.714
So we could put that there.

00:07:53.714 --> 00:07:55.989
So that counts as a voltage drop,

00:07:55.989 --> 00:08:00.072
gets a minus sign, and
it's value is R3 times i2.

00:08:04.854 --> 00:08:06.187
That's that one.

00:08:07.090 --> 00:08:09.423
Now we go over to R4.

00:08:09.423 --> 00:08:11.955
R4, this is the same
thing with two currents

00:08:11.955 --> 00:08:14.459
that we had over here with R2.

00:08:14.459 --> 00:08:16.445
So we do the same thing again.

00:08:16.445 --> 00:08:18.177
This is gonna be a... we'll define this

00:08:18.177 --> 00:08:19.677
as a voltage drop,

00:08:22.474 --> 00:08:25.225
there's the element current.

00:08:25.225 --> 00:08:28.347
And it's the same form as we had here

00:08:28.347 --> 00:08:30.766
when we calculated the current for R2.

00:08:30.766 --> 00:08:32.559
So this is gonna be a voltage drop,

00:08:32.559 --> 00:08:36.059
give it a minus sign, and it's gonna be R4

00:08:38.116 --> 00:08:41.116
times, this time it's i2 going down.

00:08:42.617 --> 00:08:43.617
i2 minus i3.

00:08:50.570 --> 00:08:52.467
And do we get home?

00:08:52.467 --> 00:08:54.398
Now we're all the way home,

00:08:54.398 --> 00:08:58.231
so our KVL equation is
all that equal to zero.

00:09:03.137 --> 00:09:05.095
So these are our two
simultaneous equations

00:09:05.095 --> 00:09:07.020
that represent our
current, and we wrote them

00:09:07.020 --> 00:09:09.090
using the mesh current method, so we get

00:09:09.090 --> 00:09:11.090
to check off step three.

00:09:11.977 --> 00:09:15.046
And in the next video, we'll
solve these two equations.


Sketching exponentials - examples

WEBVTT
Kind: captions
Language: en

00:00:01.300 --> 00:00:02.576
- [Voiceover] Now we're
gonna take the ideas

00:00:02.576 --> 00:00:04.927
from the last video and
learn how to sketch in

00:00:04.927 --> 00:00:08.240
these exponentials really rapidly.

00:00:08.240 --> 00:00:12.407
Now I gonna move this up and
we'll do a couple of examples.

00:00:16.237 --> 00:00:18.172
Here's an example circuit
I have already set up.

00:00:18.172 --> 00:00:20.766
It's an RC circuit, this is 1000 ohms,

00:00:20.766 --> 00:00:23.561
and this is two microfarad capacitor.

00:00:23.561 --> 00:00:26.527
And this voltage source
provides a step voltage to us.

00:00:26.527 --> 00:00:30.474
It's gonna start out at one
volt, and then step rapidly

00:00:30.474 --> 00:00:34.510
down to zero volts at times equals zero.

00:00:34.510 --> 00:00:38.668
And then again at ten milliseconds,

00:00:38.668 --> 00:00:41.193
it'll step up again

00:00:41.193 --> 00:00:42.925
to one volt and continue that way.

00:00:42.925 --> 00:00:46.383
This can be a clock in a
circuit, in a digital circuit,

00:00:46.383 --> 00:00:49.762
or it can be any kind of a digital signal.

00:00:49.762 --> 00:00:51.285
What we want to know, is we want

00:00:51.285 --> 00:00:53.135
to find this voltage right here.

00:00:53.135 --> 00:00:54.251
What does that look like?

00:00:54.251 --> 00:00:57.181
We're gonna use what we
know about RC waveforms

00:00:57.181 --> 00:01:00.234
to tell us what's gonna
happen here, really quickly.

00:01:00.234 --> 00:01:03.110
We don't need to use a
computer to simulate this.

00:01:03.110 --> 00:01:05.121
So, first thing I need to know,

00:01:05.121 --> 00:01:06.976
is what's the RC time constants?

00:01:06.976 --> 00:01:11.143
RC equals one thousand, one K,

00:01:12.419 --> 00:01:17.362
times two microfarads,

00:01:17.362 --> 00:01:21.237
and that equals two K times,

00:01:21.237 --> 00:01:24.705
K is plus three, micro is minus six,

00:01:24.705 --> 00:01:28.004
so this is two times
ten to the minus three

00:01:28.004 --> 00:01:31.549
or two milliseconds.

00:01:31.549 --> 00:01:33.447
So I can actually go over here to my chart

00:01:33.447 --> 00:01:37.614
and I can put a spot right
here at two milliseconds.

00:01:39.464 --> 00:01:41.570
Right there, there's five milliseconds,

00:01:41.570 --> 00:01:45.650
there's five, there's about
two and rapidly what I know,

00:01:45.650 --> 00:01:49.825
is that that line is gonna look roughly,

00:01:49.825 --> 00:01:50.938
like this.

00:01:50.938 --> 00:01:54.076
It's gonna start, of course,
it's gonna start right here

00:01:54.076 --> 00:01:56.059
and that line's gonna
go right through here.

00:01:56.059 --> 00:01:59.226
So, let's see if I can sketch that in.

00:02:00.750 --> 00:02:02.938
That's about what that's gonna look like.

00:02:02.938 --> 00:02:04.710
Now, on the other side, when it goes up,

00:02:04.710 --> 00:02:07.576
it's gonna start here

00:02:07.576 --> 00:02:09.496
and it's gonna go up to where?

00:02:09.496 --> 00:02:11.647
It's gonna go two more milliseconds.

00:02:11.647 --> 00:02:14.205
One, two.

00:02:14.205 --> 00:02:16.838
It's about two more milliseconds.

00:02:16.838 --> 00:02:19.312
That's the final value of that voltage.

00:02:19.312 --> 00:02:20.876
And I can draw another
line here like this,

00:02:20.876 --> 00:02:22.626
I can sketch that in.

00:02:24.413 --> 00:02:26.025
And the same thing's
gonna happen over here.

00:02:26.025 --> 00:02:27.836
As soon as that goes down,
I go two milliseconds over.

00:02:27.836 --> 00:02:30.222
It's gonna go right through about there.

00:02:30.222 --> 00:02:32.855
Start here.

00:02:32.855 --> 00:02:35.715
And down I go.

00:02:35.715 --> 00:02:37.616
Now, let me get a little more exact.

00:02:37.616 --> 00:02:40.797
What I want to do now is do a
little better job sketching in

00:02:40.797 --> 00:02:43.189
this exponential curve.

00:02:43.189 --> 00:02:47.353
And we know that it's about
37 percent, right about there.

00:02:47.353 --> 00:02:50.605
Here's about 37 percent

00:02:50.605 --> 00:02:54.127
of the starting voltage, right there.

00:02:54.127 --> 00:02:56.479
So that now I just use my sketching skills

00:02:56.479 --> 00:02:59.151
and I just smooth in some kind

00:02:59.151 --> 00:03:01.252
of curve that looks like that.

00:03:01.252 --> 00:03:03.932
And on the other side,
I come down 37 percent,

00:03:03.932 --> 00:03:07.516
which is roughly right about there.

00:03:07.516 --> 00:03:11.599
And I can draw, again, I
can just sketch this in.

00:03:12.440 --> 00:03:15.615
And that'll come up to some
value there, finishing value.

00:03:15.615 --> 00:03:19.282
And the same thing here,
I go 37 percent up,

00:03:20.885 --> 00:03:22.491
the exponential's gonna go through that.

00:03:22.491 --> 00:03:26.668
So it's gonna come down and
then curve off like this.

00:03:26.668 --> 00:03:28.539
And that's gonna be my estimate.

00:03:28.539 --> 00:03:30.972
Now, I didn't use a computer to do that.

00:03:30.972 --> 00:03:32.166
How do we know if we got it right?

00:03:32.166 --> 00:03:33.570
Well, I did use a computer once,

00:03:33.570 --> 00:03:35.134
so here's what it's gonna look like.

00:03:35.134 --> 00:03:38.226
Let me turn that on.

00:03:38.226 --> 00:03:41.059
And we'll move it up a little bit.

00:03:42.879 --> 00:03:45.063
And we can compare what we got.

00:03:45.063 --> 00:03:47.909
I simulated this in Excel,
just using the equations

00:03:47.909 --> 00:03:50.462
to get a really precise answer.

00:03:50.462 --> 00:03:52.557
And what you notice, is our
sketch looks pretty good,

00:03:52.557 --> 00:03:55.487
it looks pretty close to that.

00:03:55.487 --> 00:03:58.786
So what we're comparing,
is compare this, to that.

00:03:58.786 --> 00:04:01.299
And that's pretty similar, that's not bad.

00:04:01.299 --> 00:04:03.775
And that tells me that our output signal,

00:04:03.775 --> 00:04:05.260
our voltage on the capacitor,

00:04:05.260 --> 00:04:08.318
is following the digital
signal, fairly well.

00:04:08.318 --> 00:04:10.458
And it's reaching its high level.

00:04:10.458 --> 00:04:12.767
And it's rapidly coming
down as fast as it can.

00:04:12.767 --> 00:04:16.943
So, let me do one more,
let me do one more problem

00:04:16.943 --> 00:04:20.026
and we'll change the capacitor value.

00:04:25.275 --> 00:04:28.651
Now, the change here is four microferads.

00:04:28.651 --> 00:04:30.166
This is still one K.

00:04:30.166 --> 00:04:31.984
This is four microferads.

00:04:31.984 --> 00:04:33.877
So, RC, let's do it again real fast.

00:04:33.877 --> 00:04:37.977
RC equals one K

00:04:37.977 --> 00:04:41.175
times four microfarads

00:04:41.175 --> 00:04:44.008
and that equals four milliseconds.

00:04:46.693 --> 00:04:49.078
So now I know that the time
constant is four milliseconds.

00:04:49.078 --> 00:04:51.540
That means that I know
that straight line sketch,

00:04:51.540 --> 00:04:55.701
here's five milliseconds, it
goes through right about here.

00:04:55.701 --> 00:04:57.204
I start up here.

00:04:57.204 --> 00:04:59.788
I can sketch in a
straight line, like that.

00:04:59.788 --> 00:05:02.343
It goes roughly like that.

00:05:02.343 --> 00:05:03.488
Same thing over here.

00:05:03.488 --> 00:05:04.964
I can go over another four milliseconds.

00:05:04.964 --> 00:05:08.283
Up like that.

00:05:08.283 --> 00:05:12.086
And sketch in my slope.

00:05:12.086 --> 00:05:14.301
And we'll do the third one over here.

00:05:14.301 --> 00:05:16.270
Here's four milliseconds after the change,

00:05:16.270 --> 00:05:17.956
is right about there.

00:05:17.956 --> 00:05:19.802
And I sketch the line in.

00:05:19.802 --> 00:05:23.909
And now we'll go back and
do our 37 percent trick.

00:05:23.909 --> 00:05:27.368
Which is 37 percent, is right about there.

00:05:27.368 --> 00:05:30.326
And we'll come up, it
goes through right there.

00:05:30.326 --> 00:05:32.540
And if I sketch it in now,
what's gonna happen is,

00:05:32.540 --> 00:05:35.290
it's gonna go through that

00:05:35.290 --> 00:05:38.240
and something over there's gonna happen.

00:05:38.240 --> 00:05:39.919
I don't know if it's gonna finish.

00:05:39.919 --> 00:05:41.977
It might, it might not.

00:05:41.977 --> 00:05:44.351
This one's gonna go up, same way.

00:05:44.351 --> 00:05:46.076
Let's do 37 percent down.

00:05:46.076 --> 00:05:48.128
It's about there.

00:05:48.128 --> 00:05:50.006
Accurate enough.

00:05:50.006 --> 00:05:52.506
And we'll sketch in this line.

00:05:53.501 --> 00:05:55.554
And then we'll do the
same thing over here,

00:05:55.554 --> 00:05:57.972
up 37 percent.

00:05:57.972 --> 00:06:01.044
And down we go.

00:06:01.044 --> 00:06:04.454
Now, let me show you the
answer that I calculated

00:06:04.454 --> 00:06:05.805
with the computer.

00:06:05.805 --> 00:06:07.275
We'll see how that looks.

00:06:07.275 --> 00:06:08.692
Let's move it up.

00:06:12.721 --> 00:06:13.871
And we did pretty well.

00:06:13.871 --> 00:06:17.927
Notice how down here, notice
how down in this area here,

00:06:17.927 --> 00:06:20.797
we sort of got it, that's pretty accurate.

00:06:20.797 --> 00:06:22.846
I missed a little bit out here.

00:06:22.846 --> 00:06:24.374
That's okay, that's okay.

00:06:24.374 --> 00:06:27.652
But with that time constant,
we can notice right away

00:06:27.652 --> 00:06:31.452
that this waveform is
at risk of not making it

00:06:31.452 --> 00:06:33.558
all the way down to the final value.

00:06:33.558 --> 00:06:37.704
Before it's asked to turn up
and go around the other way.

00:06:37.704 --> 00:06:40.460
So that kind of intuition,
that kind of understanding

00:06:40.460 --> 00:06:43.670
of an exponential curve,
really makes you very quick.

00:06:43.670 --> 00:06:44.763
You can do this manually.

00:06:44.763 --> 00:06:46.976
And in your head, you can sketch out

00:06:46.976 --> 00:06:49.967
what these exponentials look like.

00:06:49.967 --> 00:06:51.942
The time point here, this line right here,

00:06:51.942 --> 00:06:54.161
that's RC seconds after it drops.

00:06:54.161 --> 00:06:55.928
That's the basic thing.

00:06:55.928 --> 00:06:57.943
And the other one is
this point right here.

00:06:57.943 --> 00:07:00.928
That's about 37 percent

00:07:00.928 --> 00:07:04.410
of where it started from.

00:07:04.410 --> 00:07:06.679
Thirty seven percent to go.

00:07:06.679 --> 00:07:09.466
So that's just a nice little skill to have

00:07:09.466 --> 00:07:11.522
when you're looking at exponentials,

00:07:11.522 --> 00:07:13.659
which you do all the time if
you're designing computers.

00:07:13.659 --> 00:07:15.949
Or any other kind of circuit that has

00:07:15.949 --> 00:07:19.532
sudden changes in
voltage like we had here.


Sketching exponentials

WEBVTT
Kind: captions
Language: en

00:00:01.190 --> 00:00:02.417
- [Voiceover] Now I want to show you

00:00:02.417 --> 00:00:04.975
a really useful manual skill you could use

00:00:04.975 --> 00:00:08.722
when you have voltages that
look like exponentials.

00:00:08.722 --> 00:00:10.754
And we're gonna talk
about this exponential

00:00:10.754 --> 00:00:13.137
curve here that's generated as part of the

00:00:13.137 --> 00:00:16.203
natural response of this RC circuit.

00:00:16.203 --> 00:00:19.364
And we worked out that
the voltage across here,

00:00:19.364 --> 00:00:22.447
this voltage V of T on the capacitor,

00:00:24.777 --> 00:00:28.493
its natural response is
equal to V zero, V knot

00:00:28.493 --> 00:00:31.076
times e to the minus t over RC.

00:00:35.687 --> 00:00:38.579
This value of V knot
is the starting voltage

00:00:38.579 --> 00:00:41.052
that our input source provides and then it

00:00:41.052 --> 00:00:44.862
immediately steps down
to zero and this circuit

00:00:44.862 --> 00:00:46.925
then does its natural response.

00:00:46.925 --> 00:00:49.342
We got a current coming
out of this capacitor

00:00:49.342 --> 00:00:51.189
flowing around in a circle like this.

00:00:51.189 --> 00:00:54.972
And that's the natural
response of this RC circuit.

00:00:54.972 --> 00:00:57.497
I wanna look at the
properties of this function

00:00:57.497 --> 00:00:59.902
right here, it's got some
interesting properties.

00:00:59.902 --> 00:01:01.465
I'll write it right here.

00:01:01.465 --> 00:01:04.215
V knot, e to the minus t over RC.

00:01:06.763 --> 00:01:10.196
So first thing we can look
at is this V knot value.

00:01:10.196 --> 00:01:12.137
That's this value here.

00:01:12.137 --> 00:01:13.387
This is V knot.

00:01:14.268 --> 00:01:17.508
In this particular chart, V
knot was equal to one volt.

00:01:17.508 --> 00:01:19.963
That's what it originally charged up to.

00:01:19.963 --> 00:01:22.448
And now we wanna look at
this point right here.

00:01:22.448 --> 00:01:24.844
Right when it goes
through time equals zero

00:01:24.844 --> 00:01:27.869
and the curve starts to drop.

00:01:27.869 --> 00:01:31.750
We know how high it is,
it's V knot volts high.

00:01:31.750 --> 00:01:34.574
What I wanna look at
now is what's the slope

00:01:34.574 --> 00:01:36.241
right at this point?

00:01:37.928 --> 00:01:40.991
The slope of a curve is
the derivative of a curve

00:01:40.991 --> 00:01:44.234
evaluated at this time,
at time equals zero.

00:01:44.234 --> 00:01:47.423
Let's take a derivative
of our function right here

00:01:47.423 --> 00:01:48.506
so its d, dt,

00:01:51.067 --> 00:01:53.817
V knot, e to the minus t over RC.

00:01:55.781 --> 00:01:57.614
And that equals V knot

00:01:59.202 --> 00:02:03.878
and the RC comes down, minus
one over RC comes down.

00:02:03.878 --> 00:02:07.545
Minus one over RC, e
to the minus t over RC.

00:02:11.474 --> 00:02:14.186
And that's the derivative
of this exponential

00:02:14.186 --> 00:02:15.629
for all time.

00:02:15.629 --> 00:02:17.068
So now we evaluate at

00:02:17.068 --> 00:02:18.151
t equals zero

00:02:20.682 --> 00:02:22.765
and we get V knot over RC

00:02:24.829 --> 00:02:26.412
and there's a minus sign.

00:02:26.412 --> 00:02:29.912
And it's e to the, I plug in zero for t,

00:02:29.912 --> 00:02:33.579
it's e to the zero and
e to the zero is one.

00:02:35.333 --> 00:02:37.416
So that equals the slope.

00:02:40.648 --> 00:02:42.877
At time equals zero, that equals the slope

00:02:42.877 --> 00:02:44.045
at time equals zero.

00:02:44.045 --> 00:02:47.148
So that's the value right here.

00:02:47.148 --> 00:02:49.206
That's what that slope looks like.

00:02:49.206 --> 00:02:51.927
That tangent line to the curve.

00:02:51.927 --> 00:02:54.411
Okay, now the next thing I wanna do

00:02:54.411 --> 00:02:58.361
is actually take this line
and extend it all the way down

00:02:58.361 --> 00:03:03.245
till it crosses this axis,
till it crosses the time axis.

00:03:03.245 --> 00:03:05.300
And the next question we're gonna ask

00:03:05.300 --> 00:03:07.458
is what is this point right here?

00:03:07.458 --> 00:03:09.473
What's that point?

00:03:09.473 --> 00:03:11.806
Let me move up a little bit.

00:03:13.585 --> 00:03:15.709
So now we have a line,
we've defined a line.

00:03:15.709 --> 00:03:17.846
That means we have an equation of a line.

00:03:17.846 --> 00:03:22.842
So the equation of a line
is y equals slope times x

00:03:22.842 --> 00:03:27.188
plus b, b is the intercept
on the voltage axis,

00:03:27.188 --> 00:03:28.438
m is the slope.

00:03:30.167 --> 00:03:32.964
I can plug in my values
here for our chart,

00:03:32.964 --> 00:03:34.797
y is the voltage axis.

00:03:35.896 --> 00:03:38.923
We know what the slope is,
it's sitting right here.

00:03:38.923 --> 00:03:43.006
It's minus V knot over RC
and we multiply by time

00:03:44.596 --> 00:03:49.038
and we add the y intercept
or the voltage intercept

00:03:49.038 --> 00:03:52.121
which we know is V knot, plus V knot.

00:03:53.702 --> 00:03:55.705
Alright and now what
I wanna do is find out

00:03:55.705 --> 00:03:59.416
at what time, what time
does voltage equal zero

00:03:59.416 --> 00:04:01.166
for this orange line?

00:04:02.247 --> 00:04:04.241
So we're gonna plug in zero for volts

00:04:04.241 --> 00:04:05.785
and work out what time is.

00:04:05.785 --> 00:04:07.785
Zero equals V knot times

00:04:09.786 --> 00:04:13.203
one over RC and don't lose the minus sign

00:04:15.827 --> 00:04:17.244
times t plus one.

00:04:20.843 --> 00:04:23.272
If I divide both sides by V knot,

00:04:23.272 --> 00:04:28.010
I get zero on this side
and this term on this side.

00:04:28.010 --> 00:04:31.510
So I can say zero equals minus one over RC

00:04:35.185 --> 00:04:38.435
times t plus one and we wanna isolate t

00:04:40.046 --> 00:04:42.101
so we'll take the one on the other side

00:04:42.101 --> 00:04:45.198
and multiply by RC so one more step.

00:04:45.198 --> 00:04:48.781
Minus one equals minus
one over RC times t.

00:04:52.716 --> 00:04:56.383
And in the end, we come
up with t equals RC.

00:05:05.990 --> 00:05:09.073
T equals RC is that time right there.

00:05:13.568 --> 00:05:17.813
That's how many seconds
after the step that this line

00:05:17.813 --> 00:05:19.396
hits the time axis.

00:05:21.718 --> 00:05:24.247
And notice something
here, there's no mention

00:05:24.247 --> 00:05:25.439
of V knot in here.

00:05:25.439 --> 00:05:26.642
There's no V knot.

00:05:26.642 --> 00:05:29.597
There's no V knot, it's not here.

00:05:29.597 --> 00:05:32.185
And it divided out in this step back here.

00:05:32.185 --> 00:05:34.659
It's one of our steps,
it disappeared going from

00:05:34.659 --> 00:05:37.047
this step to this step here.

00:05:37.047 --> 00:05:40.462
So independent of how
high this thing starts,

00:05:40.462 --> 00:05:42.361
it can be high or it can be low,

00:05:42.361 --> 00:05:46.322
the slope of this line
always goes right through

00:05:46.322 --> 00:05:47.905
the time equals RC.

00:05:51.121 --> 00:05:53.535
And one more thing we wanna work out is

00:05:53.535 --> 00:05:55.067
at time equals RC,

00:05:55.067 --> 00:05:56.484
what is the value

00:06:01.254 --> 00:06:02.386
of the exponential?

00:06:02.386 --> 00:06:06.219
What is this voltage
here when time equals RC?

00:06:07.380 --> 00:06:09.257
So we can use our equation again.

00:06:09.257 --> 00:06:12.464
We can plug into our equation
and find out how to do that.

00:06:12.464 --> 00:06:13.714
If I go back to

00:06:14.696 --> 00:06:18.196
V equals V knot, e to the minus t over RC,

00:06:24.351 --> 00:06:28.050
and this time we're gonna
plug in RC right here

00:06:28.050 --> 00:06:30.624
for this value here
and find out what V is.

00:06:30.624 --> 00:06:34.374
So V equals V knot, e
to the minus RC over RC

00:06:41.931 --> 00:06:45.014
or equals V knot, e to the minus one.

00:06:50.634 --> 00:06:52.220
Now the value of e

00:06:52.220 --> 00:06:54.137
is roughly equal to 2.7

00:06:55.531 --> 00:06:57.781
and the value of one over e

00:07:00.487 --> 00:07:02.737
is roughly equal to 0.37 or

00:07:04.905 --> 00:07:08.322
another way I can say that is 37 percent.

00:07:10.378 --> 00:07:13.545
So in the end, this voltage right here

00:07:15.374 --> 00:07:17.874
is about 37 percent of V knot.

00:07:21.869 --> 00:07:24.122
That's that value right there.

00:07:24.122 --> 00:07:25.661
So that's two little things we're gonna

00:07:25.661 --> 00:07:27.532
tuck away in our head.

00:07:27.532 --> 00:07:31.161
The time for that line to
hit the time axis is RC.

00:07:31.161 --> 00:07:34.083
If I wanna know the value
of where the exponential

00:07:34.083 --> 00:07:36.584
actually is at time equals RC,

00:07:36.584 --> 00:07:40.857
it's roughly 37 percent
of where it started from.

00:07:40.857 --> 00:07:43.412
Okay that's the basic idea and in the next

00:07:43.412 --> 00:07:46.622
video, I'll show you how
to use these ideas to

00:07:46.622 --> 00:07:49.062
sketch exponentials really quickly.