Dr. Alain Brizard
College Physics II (PY 211)
Electromagnetic Induction & Faraday’s Law
Textbook Reference: Chapter 29 – sections 1-7.
• Induced EMF
Oersted’s discovery that an electric current could produce a steady magnetic field can
be re-interpreted to imply that when an electric field produces a flow of current along a
wire, the current in turn produces a steady magnetic field directed at right angle to the
wire (according to the Biot-Savart law). We are, thus, naturally led to ask the following
question: Does a steady magnetic field produce an electric current and, therefore, a steady
electric field? The answer is partially yes and partially no.
Michael Faraday (1791-1867) discovered that, although a steady magnetic field does not
produce an electric current, a changing (time-dependent) magnetic field can produce an
electric current; such a current is called an induced current and is produced by an induced
emf Eind (i.e., an induced potential difference along the wire that produces an induced
electric field). Faraday conducted many experiments associated with electromagnetic induction.
• Faraday’s Law of Inductance & Lenz’s Law
The analysis of electromagnetic induction requires the concept of magnetic flux ΦB
through an open surface S (see Figure below)
defined as
ΦB =
Z
B · dA.
S
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As with the electric flux, the magnetic flux is proportional to the total number of lines
crossing the surface S. Hence, the magnetic flux through a rectangle of area A in a uniform
magnetic field B = B bz is ΦB = BA cos θ, where θ is the angle between the magnetic field
vector and the normal vector to the square surface (the SI unit for magnetic flux is the
weber: 1 Wb = 1 T · m2 = 1 V · s).
Faraday’s Law of Induction states that the induced emf in a circuit:
Eind = −
dΦB
,
dt
is equal to minus the rate of change of the magnetic flux through the circuit. The minus
sign is associated with the experimental fact that an induced emf gives rise to an induced
current whose magnetic field opposes the original change of magnetic flux (this is Lenz’s
Law).
Applications of Lenz’s Law are easily demonstrated with a wire loop (with its area
vector A pointing in the right direction) and a bar magnet (see Figures below).
In Case I (upper left box), the south pole of the bar magnet is approaching the wire loop
and, thus, the magnetic flux is becoming more negative (Φ̇B < 0). The Faraday-Lenz laws
imply that the induced current will flow in the positive direction so as to oppose this
change in magnetic flux by inducing a positive magnetic flux. The same conclusion applies
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to Case IV (lower right box), when the north pole of the bar magnet is moving away from
the wire loop and, thus, the magnetic flux is becoming less positive (Φ̇B < 0).
In Case II (upper right box), the north pole of the bar magnet is approaching the wire
loop and, thus, the magnetic flux is becoming more positive (Φ̇B > 0). The Faraday-Lenz
laws imply that the induced current will flow in the negative direction so as to oppose
this change in magnetic flux by inducing a negative magnetic flux. The same conclusion
applies to Case III (lower left box), when the south pole of the bar magnet is moving away
from the wire loop and, thus, the magnetic flux is becoming less negative (Φ̇B > 0).
Note that the magnetic flux through a circuit can change either by (1) changing the
area of the circuit, (2) changing the strength of the magnetic field through the circuit, or
(3) changing the orientation of the circuit with respect to the magnetic field, i.e.,
dΦB
=
dt
dB
dt
!
A cos θ + B
dA
dt
!
cos θ − B A
!
dθ
sin θ .
dt
• EMF Induced in a Moving Conductor
As a further application of the Faraday-Lenz laws, we consider the following problem.
A conducting rod is made to slide on a semi-infinite conducting track closed at one end
(see Figure below) in the presence of a uniform magnetic field B coming out of the page.
Here, we assume that the rod is moving at a constant velocity v = dx/dt so that the
area A = x L enclosed by the rod and the track increases at a rate of dA/dt = v L. We
further assume a resistor has been inserted into the closed circuit, with a constant electrical
resistance R.
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The motion of the rod implies that the magnetic flux through the closed circuit increases
as a function of time:
dΦB
dA
= B
= BL v > 0,
dt
dt
so that Faraday’s law states that this motion will produce a negative induced emf
Eind = −
dΦB
= − BL v < 0,
dt
which, in turn, produces an induced electric current I flowing in the circuit so as to produce
a negative magnetic flux that opposes the original increase in the magnetic flux. In the
Figure above, the induced current flows in the clockwise direction, in accordance to Lenz’s
Law.
We now note that, because a current
I =
BLv
|Eind |
=
R
R
is now flowing through the conducting rod at right angle to the magnetic field, the rod also
experiences a magnetic force
FB = (− I L by) × B bz = − BIL bx = −
!
B 2 L2
v bx
R
which slows down the motion of the rod. Hence, we keep the rod moving at constant
speed v only if an external force Fext = − FB is exerted on the rod. The external power
needed for this purpose is
Pext = Fext v = R I 2,
which equals the Ohmic power dissipated in the resistor.
• Electric Generators
An ac generator is constructed by placing a rotating wire coil (with N loops) in a
constant magnetic field. By using appropriate connections, the induced emf is
Eind = − N
dθ
ΦB
= N B A sin θ
= NBA ω sin ωt,
dt
dt
where the rotation frequency ω = dθ/dt = 2π f is also the angular frequency with which
the current flowing through the wire coil changes direction.1
1
The frequency f is 60 Hz in the US and Canada.
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• Transformers
A transformer consists of two coils known as the primary coil (with Np turns) and
the secondary coil (with Ns turns) linked together so that the magnetic flux produced by
current flowing in the primary coil passes through the secondary coil (and vice versa).
When an ac voltage Vp is applied to the primary coil, the changing magnetic field it
produces will induce an ac voltage Vs of the same frequency in the secondary coil. According
to Faraday’s Law, we find
Vs = Ns
dΦB
dt
and Vp = Np
dΦB
,
dt
so that we obtain the transfomer equation
Ns
Vs
=
.
Vp
Np
Since the electric power dissipated must be identical in each coil, Vs Is = Vp Ip, we also find
Ns
Ip
=
.
Is
Np
If Np < Ns , we have a step-up transformer since Vp < Vs , while if Np > Ns , we have a
step-down transformer since Vp > Vs . In North America, where electric power is generally
produced at locations far away from large cities, electric power is transported over long
distances at very high voltages (e.g., in excess of 100 kV) until it reaches distribution
centers in the vicinity of a large city where the voltage is progressively brought down
to 240 V outside residential homes (Note: the electric resistance of a transmission line
increases with its length).
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