


Faraday’s Law
Moving a conductor through a magnetic field is just one
way of inducing an electric current. A more common way of inducing
current, which we will examine now, is by changing the magnetic
flux through a circuit.
Magnetic Flux
The magnetic flux, , through an area, A,
is the product of the area and the magnetic field perpendicular
to it:
The A vector
is perpendicular to the area, with a magnitude equal to the area
in question. If we imagine flux graphically, it is a measure of
the number and length of flux lines passing through a certain area.
The unit of flux is the weber (Wb), where 1 Wb
= 1 T · m^{2}.
Changing Magnetic Flux
As we will see shortly, is more important than : our interest is in how flux changes, not
in its fixed value. The formula for magnetic flux suggests that
there are three ways of changing magnetic flux:
 Change the magnetic field strength: By sliding a permanent magnet back and forth, the magnetic field in a certain area will fluctuate. We will look at this phenomenon a bit later in this chapter.
 Change the area: When a bar slides on rails in a magnetic field, as in our discussion of motional emf, the square bounded by the bar and the rails gets larger. As it grows, the number of field lines passing through it increases, and thus the flux increases as the bar moves.
 Rotate the area, changing the angle between the area and the magnetic field: When the area is perpendicular to the magnetic field, the magnetic flux will simply be the product of the magnitudes of the area and the magnetic field strength. However, as you rotate the area so that it is at an angle to the magnetic field, fewer field lines will pass through it, and so the magnetic flux will decrease.
Example

First, let’s calculate the flux through the square before
it’s rotated. Because it’s perpendicular to the magnetic field,
the flux is simply the product of the area of the square and the magnetic
field strength:
Next, let’s calculate the flux through the square after
it’s rotated. Now we have to take into account the fact that the
square is at an angle of 60º:
So the change in magnetic flux is :
The magnetic flux decreases because, as the square is
rotated, fewer magnetic field lines can pass through it.
Faraday’s Law
We have seen earlier that a bar sliding along rails is
a source of induced emf. We have also seen that it is a source of
changing magnetic flux: as it moves, it changes the area bounded by
the bar and the rails. The English scientist Michael Faraday discovered
that this is no coincidence: induced emf is a measure of the change
in magnetic flux over time.
This formula is called Faraday’s Law.
Equivalence of Faraday’s Law with E =
vBl
The earlier example of a metal bar rolling along tracks
to induce a current is just a particular case of the more general
Faraday’s Law. If the bar is moving at a constant velocity v,
at which it covers a distance in a time , then:
Because is the same thing as , we get:
Lenz’s Law
Faraday’s Law tells us that a change in magnetic flux
induces a current in a loop of conducting material. However, it
doesn’t tell us in what direction that current flows. According
to Lenz’s Law, the current flows so that it opposes
the change in magnetic flux by creating its own magnetic field.
Using the righthand rule, we point our thumb in the opposite direction
of the change in magnetic flux, and the direction in which our fingers
wrap into a fist indicates the direction in which current flows.
Lenz’s Law is included in Faraday’s Law by introducing
a minus sign:
Example

We established in the previous example that the change
in flux as the square is rotated is –20 Wb. Knowing
that it takes 4 seconds to rotate the square, we can
calculate the induced emf using Lenz’s Law:
As for determining the direction of the current, we first
need to determine the direction of the change in magnetic flux.
From the diagram we saw in the previous example, we see that the
magnetic field lines, B,
move in the upward direction. Because we rotated the square so that
it is no longer perpendicular to the field lines, we decreased the
magnetic flux. Saying that the magnetic flux changed by –20 Wb is
equivalent to saying that the flux changed by 20 Wb in
the downward direction.
The direction of the current must be such that it opposes
the downward change in flux. In other words, the current must have
an “upward” direction. Point the thumb of your right hand upward
and wrap your fingers into a fist, and you will find that they curl
in a counterclockwise direction. This is the direction of the current
flow.
Conservation of Energy
Lenz’s Law is really a special case of the conservation
of energy. Consider again the bar sliding on rails. What would happen
if the induced current did not oppose the change in flux?
Since the current flows counterclockwise, the current
in the bar flows toward the top of the page. Thus, the magnetic
field exerts a leftward force on the bar, opposing the external force
driving it to the right. If the current flowed in the other direction,
the force on the bar would be to the right. The bar would accelerate,
increasing in speed and kinetic energy, without any input of external
energy. Energy would not be conserved, and we know this cannot happen.
Changing the Flux by Changing the Magnetic Field
So far, we have changed the magnetic
flux in two ways: by increasing the size of the circuit and by rotating
the circuit in a constant magnetic field. A third way is to keep
the circuit still and change the field. If a permanent magnet moves
toward a loop of wire, the magnetic field at the loop changes.
Remember that field lines come out of
the north (N) pole of a magnet. As the magnet moves closer to the
loop, the flux in the downward direction increases. By Lenz’s Law,
the current must then be in the upward direction. Using the righthand
rule, we find that the current will flow counterclockwise as viewed
from above.
As the middle of the magnet passes through the loop, the
flux decreases in the downward direction. A decrease in the magnitude
of the downward flux is the same as a change in flux in the upward
direction, so at this point the change in flux is upward, and the
current will change direction and flow clockwise.
It doesn’t matter whether the magnet or the loop is moving,
so long as one is moving relative to the other.
