Welcome back.
So by now you've heard about Faraday's law because we've had a chance to talk about it in class.
But I want to formalize some of the things that we talked about so that we can use it more generally.
You remember that Faraday's law is, at its heart, a statement about magnetic flux.
That is, I have to be able to calculate the amount of the magnetic field that's crossing some surface.
Now, we've talked about flux in a couple of different ways in this course,
we talked about it once when we were talking about Gauss's law,
where we had the electric field that was crossing a surface,
and in that case we actually had to have a closed surface
which defined the outward pointing normal of the area vector, and therefore I had no choices to make there
but I could calculate the electric flux along whatever surface that I wanted to.
Here, I'm going to be calculating today the same quantity -- the flux, but instead of the electric field
I have the magnetic field, and instead of a closed surface,
I'm going to allow myself to calculate the magnetic flux over any surface.
So I'm still interested in how much flux is -- how much magnetic field is crossing that surface.
That is, I'm integrating the field strength in the direction of the normal to the surface dA,
and the information about the direction of the surface
in the direction of the field is contained in the fact that I'm taking the dot product
between the magnetic field and the surface.
So if I have a general magnetic field
and I have some surface, I always have to worry about the relative orientation
of the area vector, which parameterizes the direction of the surface
and the magnetic field, in order to really find out how much of that field is traveling through the surface.
And now, unlike in Gauss's law, I'm going to allow myself to have any surface
and actually it's most -- this is most useful if I'm talking about like a ring of, you know, a conducting ring
where I'm allowed to have currents flow
so my surface is in fact never going to be closed, and therefore
I have a choice that I have to make about the direction of the normal vector.
I don't have an outward direction anymore.
And in this circumstance, you can choose whichever direction for the normal you want
but once you choose that direction for the normal, you are
injecting information into the mathematical model,
and that choice of the direction of the normal then, is going to help you to figure out the
sign of the direction of the induced currents later on.
So when I choose the direction of the normal for my surface
I'm making a choice as to what's positive and what's negative
and that choice is going to carry through the rest of the problem with me.
So when you're drawing your diagrams for problems using
magnetic flux, you always want to be very clear about the choice that you're making
for the direction of the normal to the surface.
Because the sign of that direction really matters very much.
And once I've calculated that magnetic flux, and usually my surface is bounded,
as I said, by some conductor like a wire, right,
I'm going to then be able to calculate things that -- I'm gonna be able to calculate
properties along the edge of that surface, and my direction
along the edge of that surface has now been chosen along with your choice of the normal vector.
So if you're choosing a normal vector to be in some direction, say upward,
then I use the right hand rule to tell me that my choice
for positive direction is going to be counterclockwise.
And so all of my choices are consistent with each other
if I have a normal vector pointing in one direction
I've said that the counterclockwise direction with respect to that normal is gonna refer to positive things.
And the next step, as you'll remember, is once
I've been able to calculate the flux,
nothing really interesting happens until that flux starts to change, right?
So once I have a magnetic flux that's dynamic,
once I have a magnetic flux that's either increasing or decreasing,
that magnetic flux is going to induce a current along the boundary to my surface,
which is, of course, in general, is a wire.
So changing the magnetic flux leads to an induced EMF
which leads to an induced current along the boundary of the surface that I'm looking at.
And how you want to remember this, that is, how do you want to remember the fact
that changing magnetic flux will lead to induced currents,
is this idea that magnetic flux hates change.
When I start to change the magnetic flux
the conducting surface that surrounds my area,
is going to induce a current that tries to fight the change that you're trying to make.
So in a conceptual sense
it's this -- Faraday's law tells us that the
that the current that's flowing around my surface is going to act to
counteract
the change in magnetic flux that I'm experiencing across the area.
And what you know about generating magnetic fields then, can help you to understand
why the current along the boundary of my surface is traveling in one direction or the other.
Nature hates changing magnetic flux
and it's going to induce currents along the boundary that are going to fight
fight that change in magnetic flux.
Very
Very rarely does it ever win, but it's going to do its best to fight it.
Okay, so to make this a little bit more mathematically concrete
let's think of an actual example. So here, I have a current
that's in the plane of the page, or I have a -- I'm sorry, I don't have a current,
I have a loop of conductor that's in the plane of the page, and I have a magnetic field
that's heading into the page.
So I can calculate the magnetic flux here, because I have a magnetic field that's crossing my surface
and because of the way that I've oriented my surface, I know that the normal vector, no matter which direction
I choose, is going to be either parallel or antiparallel with the magnetic field, right?
So that their directions are going to line up.
Well, I can either choose my area vector to point in the direction of the magnetic field, also into the page,
or I could choose my normal vector to point out of the page.
I think to make this sort of
maximally interesting
I'm going to choose my normal vector to point out of the page, right, so that way my
normal vector and my magnetic field are actually antiparallel to each other.
So if my area -- if my total loop has some area that I'm gonna say is just equal to A,
then I can start to calculate the magnetic flux here
because it's the integral of B dot dA.
Well, in this case my area is a constant and my magnetic field is a constant,
and at least for -- my magnetic field is a constant value across the page,
and so I don't actually have to integrate at all because
the value of B is independent of location and therefore the integral
collapses just to the magnitude of B times the area of A.
So I'm gonna write this as the magnitude of B times
the -- the magnitude of A, but of course, there's this cosine theta, right? There's a cosine of the angle
between my magnetic field and my area vector. This theta here is our standard
definition of theta when I'm doing a dot product, is the area between the two vectors.
And so in this example
the angle between the two vectors is 180 degrees
meaning that the cosine of 180 degrees is minus 1
and I'm left with minus the magnitude of the magnetic field
times the area vector A -- times the area of the loop.
And now this number is negative, and now you you might have a little bit of discomfort with this because
you know, we've generally dealt with fluxes that turn out to be positive
except of course in Gauss's law when I had an enclosed negative charge,
but remember, this minus sign comes from a choice that I've made about the direction
of the normal vector to the surface.
But what it means is that if I change the flux
then the sign that I get from doing that derivative
is going to correspond to the direction of the current around the loop, right?
Because I've chosen A to be upward, or towards me in this case, if I end up with a with an
induced current or an induced EMF that's positive, that means
I'm going to be dealing with counterclockwise current in this loop.
If I'm dealing with -- if I get an induced EMF
that's negative, that means I'm going to be talking about clockwise current in the loop.
So the right hand rule tells me now about the direction of the boundary
because of a choice that I've made about the direction of the area vector.
I really defined my coordinate system, right? That's another word for it.
So if I let -- if I try to calculate the change in flux here, if I try to do d flux dt,
I have to allow one of these things to change.
The sort of obvious one here is the strength of the magnetic field, right?
So if I keep the area constant, and I keep the orientation constant,
then I end up with minus A times the derivative of the change
of the strength of the magnetic field with respect to time.
Now I have to make a choice. Let's let dB dt be greater than 0.
So if the magnetic field is getting stronger,
that means that the change in flux with respect to time is going to be minus A
times something
positive, right? And so d -- the change in flux with respect to time is actually going to be negative.
So if I plug that into Faraday's law as we saw it in class, the induced EMF is minus d flux dt, right?
It's going to fight the change in magnetic flux, and therefore it's gonna have that minus sign.
It's actually going to be greater than 0.
Right? Because the thing that -- the d flux dt
is less than 0, I multiply that by minus 1, and I get something that's greater than 0.
So in this example
what I see is that the changed --  the induced EMF,
which induces a current, is going to be positive, which means that the induced current is going to be
counterclockwise around this circle.
If I had done all the signs backwards
I would have ended up with a different direction of the induced EMF
and therefore a different direction of the current.
Mathematically, I would have ended up with a different sign for the induced EMF
but in physical reality
I would have ended up with the same direction.
So it's important to be very clear about the choices that you make
so that when I'm grading your problem set or exam, I know that your final
mathematical answer is consistent with actually physically what's happening in the problem.
So now we can also see this makes sense to us because the
magnetic field that's going to be created by this current, if I grab the top of the wire
it's going to create a magnetic field inside the wire that's coming towards me. Well, of course
that's gonna fight the increased magnetic field going away from me, right? So the induced magnetic field
that's going to be created by the induced current
is going to fight the change of the external magnetic field.
And that brings me full circle,
unintended, to where I started in this problem
in that nature is somehow trying to block the change in magnetic flux
by creating this current along the boundary.
Okay, I want to do one more interesting example.
So consider the case where I have an external magnetic field
that's going to be pointing directly upward
and I have a
a loop in that external magnetic field
and instead of allowing the
strength of the magnetic field to change, or the area to change
I'm actually going to allow the orientation between the area and the magnetic field to change.
And so I can think about that as sort of putting my loop on a stick and allowing that stick
allowing the stick to rotate
around this axis with some angular velocity, omega.
So if I go to my expression for the magnetic flux, I see that the magnetic flux --
now, the integral isn't going to -- is going to be trivial here
because the area is constant and the magnetic field is constant.
So the only thing that's, you know, the only thing that's dynamic here
is theta, which doesn't appear in the integral,
so I'm going to end up with it being the magnitude of B
times the area of the loop times the cosine of the angle between them.
And now, this can be kind of interesting if I go back to the left hand side here
because theta, the angle between them,
is parameterized by an angular velocity times time.
So if I try to calculate the EMF, which is minus d flux dt,
I get that the time derivative is gonna affect
the only time dependent thing in the problem
which is theta, and so I end up with the magnitude of B times the magnitude of A
times d by dt of
cosine omega t.
Well the derivative of cosine omega t is
actually -- I can't forget my minus sign, right? --
the cosine of omega t is actually pretty easy. It's actually minus sine omega t times omega,
So I end up with plus the magnitude of B times
A times sine omega
t times omega.
And so I get something really interesting. As my loop is rotating through my magnetic field
the induced EMF is actually going to not only change in magnitude
and that magnitude is going to depend on both B, A, and the rate at which you're spinning it,
but it's also going to have this transcendental property
where it's going to be changing sign as it's rotating.
Now physically you can think about this as, as the loop is rotating
sometimes the flux is going to be getting bigger and then smaller
and then more negative and then less negative, and so as you
try to fight that change
the induced current is also going to have a similar pattern.
Now, those two strengths are going to be out of phase
because you're rotating it with a cosine and the response is a sine, right? But that's again
this statement that you're fighting the change, but you don't always win.
But the induced EMF, and hence the induced current on this loop, is going to have an oscillatory behavior.
Right? Now this should remind us kind of our discussions
about what is the torque on a on a current loop in an external magnetic field?
It's just sort of the opposite problem
where if I have a constant magnetic field and I try to spin a loop in it
that action of me trying to spin the loop in the external magnetic field
is going to generate a current on that loop that has
that has an oscillating property. It's gonna get stronger and less strong.
And of course, now
this is the reason why it's so natural to have
to have AC current, right, that's generated in a power plant where a turbine is spinning, right?
Turbines take, you know, take rotational energy, and they're able to then rotate
current -- they'll rotate loops in a magnetic field
that's set up by some permanent magnet
and then once you do that you're able to generate a current or generate an EMF
that has a time dependence, and that time dependence is a transcendental function.
So don't forget the three different ways you can change the magnetic flux, right?
You can change the magnetic field,
you can change the area vector, or you can change the angle between them,
and all of these methods are pretty powerful at changing
changing the magnetic flux, and therefore being places where you would use Faraday's law.
Okay, so the take-home messages are these.
First, as we talked about in class, as we talked about during this video, changing a magnetic flux
induces an EMF.
It's the change of the physical property of the magnetic flux that leads us
to having currents induced on the boundary of those surfaces.
Change happens in all sorts of ways. You can change B, you can change A, you can change B and A,
you can change the angle between them, but you have to keep track of what's changing.
And then the mathematical model, that is the sign that you get out of the EMF by using Faraday's law
comes from your choice
of the normal vector to -- of the normal vector to the surface that you're studying.
