Electromagnetism
G. L. Pollack and D. R. Stump
10. Electromagnetic Induction
Chapter Summary
Faraday’s law If a magnetic field changes in time, there is an induced
electric field. The field equation that describes the phenomenon of
electromagnetic induction is
∇× E = −
∂B
,
∂t
which is called Faraday’s law. The integral form is
I
dΦ
E=
E · d` = −
dt
C
where Φ is the magnetic flux through any surface with boundary curve
C.
Lenz’s law The direction of E in electromagnetic induction opposes
the change of flux; i.e., if a conductor is present then the induced current
produces a magnetic field in the direction that tends to maintain the
flux.
Self-inductance. A current I in a conducting loop creates a magnetic field. The flux through the loop is proportional to the current,
Φ = LI. The constant of proportionality L is the self-inductance, which
depends on the geometry of the conducting loop. If I changes in time,
there is an induced emf around the loop, and by Faraday’s law,
E = −L
Mutual inductance
dI
.
dt
For 2 loops,
Φ12 = M I2
and Φ21 = M I1 .
By Faraday’s law, the emfs induced around the loops are
E1 = −L1 I˙1 − M I˙2
E2 = −L2 I˙2 − M I˙1
The energy density of the magnetic field is umag =
B2
.
2µ0
Motional emf If a conducting loop moves in a static magnetic field,
the emf around the loop is E = −dΦ/dt.