>> Faraday's law says that if
you have a changing B field,
or if through a closed area,
if you have the flux changing
as a function of time, it
is going to induce some EMF.
Something like a
voltage along that loop.
Also, if you have changing
magnetic field, you're going
to create E field, and this
kind of E field is different
from the E field
created by charges.
This kind of E field,
they can form loops.
So, if you integrated those
induced E field along a closed
loop, you're going to
get null zero value,
if the B field is
changing in that area.
And then, the flux through
the region is going to change
as a function of time and the
time derivative is not zero.
These two equations
essentially they're talking
about the same thing.
This does not rely on a real
physical circuit or loop.
And it describes the B E field,
which can be induced whenever
your B field is changing.
And this is the same
idea, but it relies
on physical, actual circuit.
And the circuit can have
actually not only one turn,
it can have more than one.
It can have N turns.
The total accumulated
EMF induced
in the circuit is equal
to N times of that.
And the flux, again, the flux
is defined to be the number
of magnetic field lines
penetrating a certain area.
And that is equal to B,
the B field dot product
of a normal reaction
of that surface,
and then integrating
small patches of A.
If the B field is uniform,
it may even still have
an angle comparing
to the normal direction
of the area.
Still, if the B field is
uniform, we can take it
out of the integral and this
flux calculation becomes simply
the B field if your
surface area is flat,
then it became B
field multiply area.
And the angle between
the uniform B field
and the flat surface, the--
it's going to be fixed.
And we choose the angle to be
the angle between the B field
and the normal direction
of the surface.
And that's going to
give you a cosine sigma.
If you have for B
field into the page,
and you have your surface
parallel to the page,
and then a normal
direction and the B field are
in the same direction, the flux
equal to B multiply that area,
multiply cosine zero,
which is 1.
If your B field is into the
page and your area is like this,
a normal direction, and
the B field direction are
perpendicular to each other,
in that case cosine 90 degree
give you zero, you no flux,
if your B field is into the page
and your surface is
parallel to the B field.
Or, if you're at an angle,
then basically you just
use the cosine of the angle
between your B field
and the normal direction
of your surface.
And that's the angle
sigma, here.
So, let's use two examples
to apply Faraday's law.
First is that if in a circular
region we have uniform B field,
however this B field is
changing as a function of time.
This can be any type of a
format, sine, cosine, linear,
increasing or decreasing
with time,
or any other function of time.
Anyways. It's a B field
changing as function of time,
and we know that's going
to induce some E field.
And the E field is going to
be looping and at location P,
we try to find out
the induced E field.
And because this region
is circularly symmetric,
we can imagine choosing a
circular loop for Faraday's law,
and this circular loop
should pass the radius R,
equals to the distance from
the center to the location P
and then this is your loop.
Okay? Along this loop, we
can figure out the E field.
And also, notice
this negative sign.
The negative sign says
that those induced E field,
or induced EMF, always try
to go against the change.
It does not like change.
So. Now if the B
field increases,
this function is a B
field increases with time.
And then, the flux going through
this region, flux is equal
to the B multiply area.
Now the B is perpendicular
to the area.
B is parallel to the normal
direction of the area.
Cosine 0 equal to 1.
So simply flux equal to that.
If B is increasing with
time, area is not changing.
We know phi B is
also increasing.
If phi B is increasing
and also into the page,
then we know the induced B
field will go against the data.
So the induced B field
will be opposite.
Will be out of page.
So, what can give you an
induced B field out of page?
Imagine current flow in
what direction is going
to give you induced
B field out of page?
Using our right-handed
rule number 3,
we see that if our
induced E field,
or you can imagine current
flowing counterclockwise,
this way, we are going
to get induced B field,
canceling out the
increasing initial B field.
That means the induced
E or current.
I put a quotation mark here
because we do not
really have a current.
We do not have really
have wires.
Imagine if we have a wire,
that's what a current
is going to be.
But that's a direction
of the induced E field.
It's going to be
counterclockwise.
Okay? And then we can further
solve this problem by looking
at this equation and then for
this loop we have the E field ds
because the E field
and the ds is the same
direction everywhere.
This integral for the closed
loop gave us E multiply
integration of the total amount
of ds, which is 2 pi r. Okay?
And that's equal to d phi B dt.
Which is negative d phi
is B and the area dt.
And this area is simply not
changing as a function of time.
So you can take area out.
That's negative sign.
dB, dt. Or so, the area
is equal to pi r square.
Now, the amount of area
which provides us the
flux is the same area,
like has the same radius
like this integration loop.
So they have the same R, so
this is the E field multiply the
circumference equals
to the negative sign
of the area multiply dB dt.
If you know the function
of B as a function of--
if you know B as a function
of t, you can find the dB dt,
and then you can solve
for the induced E field
and you know that's in the
counterclockwise direction.
Similarly, we can also
do something here.
If we have a conducting
wire arranged this way,
and then we have another
conducting wire touching that,
but moving with a elastic,
constant elastic V
in that direction.
What's going to happen is
that's let's look at the flux.
The flux actually B
equals to B multiply area.
Uniform B field into the
page, multiply this area.
And if this is zero,
this is location X
and that's moving
toward the X direction.
Okay? So, and if this
is the length is L,
so it's B multiply area, and the
cosine zero degree give you 1.
The total flux, it's that
without any extra angle.
And that's equal to B
multiply L multiply x.
So now let's take
at the dB, d phi dt.
How does flux change
as function of time?
Which is, d B L X. B is not
changing, is uniform constant.
So here we have a constant
B. But what is changing?
Because this wire is looping.
The area of this
loop is changing.
This constant L is constant.
So, the only thing changing is
x. So we have B L and dx dt,
which is B L multiply
this V. We lost it.
Okay? That's d phi B dt.
And that's going to
be equal to the amount
of EMF induced inside.
The absolute value
of that equals
to d phi B dt absolute value,
which is equal to B L V.
But in which direction?
Because this area is increasing,
we are having more
flux into the page.
And induced B field is
going to be out of page.
The induced current is going
to be counterclockwise,
like we just did before.
So we have a current going this
way and this way and this way.
So, as if we are
generating the battery,
which is in this direction,
which has battery voltage equal
to B L V. And that's
in a circuit.
If you're here, you
have a resistor,
and then you could
generate a current.
Okay? And also, if you
have a resistor, R,
and then you have
the epsilon over--
We are going to have the current
I equals to the epsilon over R
if you have a resistor here.
And then correspondingly, you
know, the power generated here
or used here is equal to I,
or epsilon, which is equal
to epsilon square over
R, which is determined
by the B field strengths, the
lengths of the wire, the speed
and that square over
the resistance.
And also, when the energy is
consumed inside this wire,
somebody pulling this
wire to move this way
to maintain a constant
speed has to do some work.
Okay? And that's the
examples of Faraday's law.
Basically the idea is
that whenever you have
a changing B field,
even though your area and
nothing else, that's not change,
or even if you do not really
have a loop of circuit,
you are going to generate
an E field which is looping
and that E field can be
calculated using this equation.
While you choose the appropriate
loop and simplify this integral
and then calculate the
derivative of flux versus time.
And if you do have
a real circuit,
and then you can change the
flux going through that loop,
by either changing
the area, like this,
or you can change the B field,
make it stronger or lower
by controlling the
external B field,
or you can also change
the angle set
that you have a fixed B
field, you have a fixed loop,
but if by rotating it inside the
B field, you can change the flux
and then your d phi
dt is not zero,
you induce some voltage
in your loop.
And that's exactly how the
electricity that we are using
in the city are generated.
