Faraday’s Law of Electromagnetic Induction
EE 141 Lecture Notes
Topic 20
Professor K. E. Oughstun
School of Engineering
College of Engineering & Mathematical Sciences
University of Vermont
2014
Motivation
Faraday’s Law of Electromagnetic Induction
Consider a simple closed circuit C enclosing an open surface S in a
magnetic field B(r). The magnetic flux Φm linking the circuit C is
defined as
ZZ
Φm ≡
B · n̂d 2 r
(1)
S
where n̂ is the unit normal vector to S in the chosen positive sense.
Faraday’s law of electromagnetic induction (1831) then states that
the induced electromotive force (emf)
I
E ≡ E · d ~ℓ
(2)
C
in the positive sense about the closed circuit C (as determined by the
choice of n̂) is related to the change in magnetic flux through the
circuit by
d Φm
E=−
(3)
dt
Faraday’s Law & Lenz’s Law
This electromotive force acts as the driving force for the current
induced in the closed circuit C, and Faraday’s law states that this emf
results from any change in the magnetic flux linking that circuit.
Notice that the mathematical expression (3) of Faraday’s law is a
statement of an independent experimental law and is entirely
independent of the manner in which the magnetic flux linkage
changes. Nevertheless, due care must be taken in defining the
respective reference frames that contain the circuit and the magnetic
flux source, as is the case in the so-called Faraday’s paradox.
The negative sign appearing in Faraday’s law indicates that the
induced emf opposes the change that is producing it. This then leads
to Lenz’s law which states that a change in magnetic flux induces an
emf that produces a current whose magnetic field opposes the
original change in magnetic flux.
Faraday’s Paradox
Faraday’s paradox results from an experimental arrangement for
Faraday’s law that consists of a conducting cylindrical disc and a
coaxial cylindrical permanent magnet that are separated by a small
distance along their common axis. The magnet’s B-field is then
directed along this axis. In order to measure the effects of this field,
an electric circuit is formed through a pair of sliding contacts situated
at the center and the rim of the conducting disc.
When the disc is rotated with angular frequency ω and the
magnet is held fixed, the magnetic Lorentz force Fm = qv × B
produces a radial current that is outwardly directed in the
conducting disc, resulting in a measured direct current in the
circuit. In this arrangement the device is known as a Faraday
generator or Faraday disc.
If the disc is held fixed while the magnet is rotated about its
axis, no current is measured.
If both disc & magnet are rotated together, current is measured.
Faraday’s Paradox
This experiment is described as a paradox because the results appear
to violate Faraday’s law as the magnetic flux linkage with the
conducting disk appears to be the same regardless of which element
is rotating so that d Φm /dt = 0 and there should never be an induced
emf or current flow.
Alternatively, from the viewpoint of the magnetic lines of flux, the
induced emf is proportional to the rate at which the magnetic flux
lines are cut. If the flux lines originate in the magnet, then they are
stationary in the frame of reference of the magnet so that rotating
the disc relative to the magnet, either by rotating the disc or the
magnet, should then produce an emf, but rotating both of them
together should not.
The resolution of this apparent paradox is obtained simply by noting
that the motion of the magnet is entirely irrelevant in Faraday’s law.
Self-Inductance
From the Biot-Savart law (18.3), the magnetic flux linking an isolated
circuit is dependent on the circuit geometry and is linearly related to
the current in the circuit1 . For a stationary circuit with fixed shape,
the only change in its magnetic flux linkage must then result from
changes in the current, so that d Φm /dt = (d Φm /dI )(dI /dt).
Because Φm is directly proportional to the current I , then d Φm /dI is
independent of I . This quantity is defined as the inductance L, where
L≡
d Φm
dI
(H)
(4)
With this identification, Faraday’s law (3) becomes
E = −L
dI
,
dt
where the MKSA unit of inductance is the henry (H ≡ Vs/A).
1
If a ferromagnetic material is used, then the relation between Φm and I is nonlinear and dΦm /dI is no longer
independent of I . In that case dΦm /dI is called the incremental inductance and Φm /I the inductance.
(5)
Mutual Inductance
Consider a system of n isolated closed circuits Cj , j = 1, 2, . . . , n,
each with current Ij . The magnetic flux Φmi linking the i th circuit is
then given by the sum of the individual flux linkages produced by the
currents in each of the n circuits, so that
Φmi =
n
X
Φmij ,
(6)
j=1
where Φmij is the magnetic flux linking the circuit Ci due to the
current Ij flowing in the circuit Cj . The induced emf in the closed
circuit Ci is then given by Faraday’s law as
n
X d Φmij
d Φmi
Ei = −
=−
.
dt
dt
j=1
(7)
Mutual Inductance
If each circuit Cj is rigid and stationary in the laboratory reference
frame, then the only change in each magnetic flux linkage Φmij is that
which is caused by changes in the currents Ij , so that
d Φmij /dt = (d Φmij /dIj )(dIj /dt),
where d Φmij /dIj is independent of the current Ij in the linear case.
This quantity then defines the mutual inductance
Mij ≡
d Φmij
,
dIj
i 6= j
(H)
(8)
between circuit i and circuit j. Notice that this expression reduces to
the self-inductance of the i th circuit when i = j, where Li = Mii .
Mutual Inductance
The mutual inductance for a pair of stationary, rigid circuits in a
linear magnetic medium is given by M21 = d Φm21 /dI1 = Φm21 /I1 .
The magnetic flux Φm21 is obtained from the Biot-Savart law (18.3)
and Eq. (20.1) as
)
Z Z (I
d ~ℓ1 × (r2 − r1 )
µ
I1
· n̂dr22 .
(9)
Φm21 =
3
4π
|r2 − r1 |
S2
C1
Because
I
d ~ℓ1 × (r2 − r1 )
d ~ℓ1
=
∇
×
,
2
|r2 − r1 |3
C1
C1 |r2 − r1 |
the mutual inductance M21 linking circuit 2 to the current flowing in
circuit 1 is given by
(I
)
ZZ
µ
d ~ℓ1
M21 =
· n̂dr22 .
∇2 ×
4π
|r
−
r
|
2
1
C1
S2
I
Mutual Inductance
Application of Stokes’ theorem to the surface integral in this
expression results in Neumann’s formula for the mutual inducatnce
M21
µ
=
4π
I I
C2
C1
d ~ℓ1 · d ~ℓ2
|r2 − r1 |
(10)
The inherent symmetry of this expression then shows that, in general
Mij = Mji .
(11)
Notice that, in principle, Neumann’s formula is applicable to
determining the self-inductance of a circuit. However, due care must
be given to the logarithmic singularity at r2 = r1 in the resultant
contour integral. This singular behavior may be resolved by including
the thickness of the current-carrying circuit, a quantity which, in
most cases, may be conveniently neglected in determining the mutual
inductance.