24
2 Basic Electromagnetism
introduce superconductors, in which current causes magnetic behavior, as one of
magnetic substances in such a way as to make electricity and magnetism more equal
to each other, which will contribute to advancing the E-B analogy.
2.4 Electromagnetic Phenomena Varying with Time
When electromagnetic fields vary with time, the equations described in Sects. 2.1
and 2.2 do not hold. One of them is the electromagnetic induction. Faraday’s law
describes how the electromotive force is induced by variation in the magnetic flux
with time. That is, if the magnetic flux that penetrates a closed circuit C is denoted
by , the electromotive force is given by
V em = −
d
dt
,
(2.46)
where the directions of the magnetic flux and electromotive force obey the right-hand
rule. The left side of (2.46) is given by the curvilinear integral of the induced electric
field. Hence, we have
C
E · ds = −
d
dt
S
B · dS.
(2.47)
Using Stokes’ theorem, the left side of this equation is rewritten as the left side of
(2.17). Since the plane S does not change with time, the order of the time derivative
and surface integral can be changed:
S
(∇ × E) · dS = −
S
∂B
∂t
· dS.
(2.48)
Since this holds for arbitrary S, we have
∇ × E = −
∂B
∂t
.
(2.49)
This is the differential form of Faraday’s law. When there is no variation with time,
this equation leads to (2.19), which describes the electrostatic field. Hence (2.49)
holds generally, including the electrostatic field.
Assume that plane S moves with velocity V in a space with uniform magnetic flux
density B. We calculate the magnetic flux that enters S during a short period t.
Since the magnetic flux that enters through a small segment ds, which is the magnetic
flux in the hatched region shown in Fig. 2.9, is (Vt × ds) · B = (B × V) · dst,
is obtained as
2 Basic Electromagnetism
introduce superconductors, in which current causes magnetic behavior, as one of
magnetic substances in such a way as to make electricity and magnetism more equal
to each other, which will contribute to advancing the E-B analogy.
2.4 Electromagnetic Phenomena Varying with Time
When electromagnetic fields vary with time, the equations described in Sects. 2.1
and 2.2 do not hold. One of them is the electromagnetic induction. Faraday’s law
describes how the electromotive force is induced by variation in the magnetic flux
with time. That is, if the magnetic flux that penetrates a closed circuit C is denoted
by , the electromotive force is given by
V em = −
d
dt
,
(2.46)
where the directions of the magnetic flux and electromotive force obey the right-hand
rule. The left side of (2.46) is given by the curvilinear integral of the induced electric
field. Hence, we have
C
E · ds = −
d
dt
S
B · dS.
(2.47)
Using Stokes’ theorem, the left side of this equation is rewritten as the left side of
(2.17). Since the plane S does not change with time, the order of the time derivative
and surface integral can be changed:
S
(∇ × E) · dS = −
S
∂B
∂t
· dS.
(2.48)
Since this holds for arbitrary S, we have
∇ × E = −
∂B
∂t
.
(2.49)
This is the differential form of Faraday’s law. When there is no variation with time,
this equation leads to (2.19), which describes the electrostatic field. Hence (2.49)
holds generally, including the electrostatic field.
Assume that plane S moves with velocity V in a space with uniform magnetic flux
density B. We calculate the magnetic flux that enters S during a short period t.
Since the magnetic flux that enters through a small segment ds, which is the magnetic
flux in the hatched region shown in Fig. 2.9, is (Vt × ds) · B = (B × V) · dst,
is obtained as