Faraday’s Law of Induction
What happens when we have a changing electric current in a circuit? We have seen that
an electric current produces a magnetic field, so a changing electric current will produce a
changing magnetic field. Are there any further consequences of this? Yes, indeed. A changing
magnetic field will produce an electric field as we will learn in the following discussion.
Suppose that we have a loop of wire which forms a closed circuit, but which has no
battery or other source of emf in the circuit. Up until now in our studies, no matter what we
might have done with this loop, there would be no current flowing in its wire since there was no
emf to produce an electric field within the wire to push electric charges around making up a
current. However, it is found experimentally that if a changing magnetic field is created in the
region of the loop, that an “induced” current will flow in the loop even though there are no
sources of emf (battery or power supply) connected in the circuit. We say that the “induced”
current is created by an induced emf, which itself is created by the changing magnetic field at the
loop.
Faraday discovered that the induced emf is directly proportional to the time rate of
change of the magnetic flux through the circuit loop. We can write Faraday’s law of induction as
dΦB
ε =−
,
dt
where ΦB is the magnetic flux through the circuit loop, which we defined in the last class as
G G
Φ B = ∫∫ B ⋅ dA ,
S
where S is the area of the circuit loop. If the loop consists of N turns, then the induced emf is
given by
dΦB
ε = −N
.
dt
We will discuss the significance of the negative sign in a moment.
If the magnetic field is uniform, then the magnetic flux can be written more simply as
ΦB = BA cos θ, so that Faraday’s law becomes
d ( BA cos θ )
ε =−
.
dt
From this expression, we see that an induced emf can be created in 3 basic ways:
1. the magnetic field can vary with time
2. the area of the circuit loop can vary with time
3. the angle θ, between B and the normal to the loop, can vary with time.
Of course, any combination of these can also produce an emf.
Let’s discuss examples of how to produce an emf using each of the 3 methods listed
above.
Example 1:
Changing B – moving magnet towards or away from a fixed solenoid coil – demo
Example 2:
Changing A – changing the area of a coil within a constant magnetic field – demo.
Example 3:
Changing θ – rotating a coil in a uniform B field – an electric generator – demo.
Example 4:
Changing B – but more complex demo –
Time-varying electric current in one circuit (from a radio or audio receiver) producing a
time-varying B; the time-varying magnetic flux “picked up” by an isolated solenoid connected to
a speaker can then play back the sounds with no direct connection to the speaker.
Note that an emf can be written as
G G
ε = ∫ E ⋅ ds
just as any potential difference can be written. However, over the closed loop circuit, we have
that
G G
ε = v∫ E ⋅ ds .
Normally, for conservative vector fields this line integral would be zero, however it is non-zero
in this case, implying that the electric field created by an induced emf is non-conservative.
Faraday’s law can be written as
G G
dΦ
d  G G
vC∫ E ⋅ ds = − dt B = − dt  ∫∫S B ⋅ dA
where S is any surface bounding C. We will return to this in a few days.
Now, next let’s discuss that minus sign in the equation. Why is it there and what is it
telling us? The minus sign is actually short for Lenz’s law – a statement about conservation of
energy that is appropriate to Faraday’s law. It is best to discuss this through examples.