3.3 Merits of Introducing Superconductivity
43
conductor for which Ohm’s law holds is used, such a situation cannot be realized.
The width, length, and thickness of the transmission line are denoted by w, l, and
d , respectively. Since the magnetic flux density B is common to the two magnetic
materials, the surface current densities are τ 1 = B/μ 1 and τ 2 = B/μ 2 . Hence the
total current is
I =
w
2
(τ 1 + τ 2 ) =
(μ 1 + μ 2 )wB
2μ 1 μ 2
.
(3.27)
Since the magnetic flux is Φ = dlB, the self-inductance, i.e., the magnetic flux stored
by a unit current, is
L =
Φ
I
=
2μ 1 μ 2 dl
(μ 1 + μ 2 )w
.
(3.28)
From the viewpoint of impedance, 1/C corresponds to L. Since corresponds to 1/μ
from (2.2) and (2.22) (note that μ has dimension of (s/m)
2 ), and S and dl are the
areas of the planes normal to E and B, respectively, it can be understood that (3.26)
and (3.28) are analogous to each other. Such effectiveness in education is also one
of the merits of introducing superconductivity.
(4) Derivation of magnetic energy and prediction of the induction law
One of other big merits of introducing superconductivity into electromagnetism is
direct derivation of magnetic energy from mechanical work. The electric energy
can be calculated from the mechanical work needed to carry small electric charges
from infinity until the final desired distribution of electric charge is completed. If
we similarly try to derive the magnetic energy, however, the attractive force works
between currents, and the mechanical work to carry a current is negative. This is
attributed to the electromotive induction. For this reason, it is common that students
learn the magnetic energy, which is derived through equivalent electric energy, after
learning about electromotive induction. That is, learning about dynamic phenomena
is required to learn about the static magnetic phenomena.
On the other hand, it is possible to directly derive the magnetic energy from the
mechanical work done by the Lorentz force in a circuit composed of superconductors.
Suppose that there is a closed circuit made of superconducting slabs, as shown in
Fig. 3.14. The superconducting circuit is composed of a fixed superconducting slab
with three sides and a movable superconducting slab with superconducting contact
between them. For simplicity, b is assumed to be much larger than a. It is assumed
that the magnetic flux is stored in the space surrounded by the circuit. This situation
can be achieved by the following procedure: external magnetic field is first applied
to the superconducting circuit at a temperature higher than the critical temperature,
and then the temperature is lowered below the critical temperature, which makes
the circuit superconducting. Finally, the external magnetic field is removed. This is
called the field-cooled process. In this situation, a shielding current is induced in
the circuit. This process is employed to realize a permanent current. The Lorentz
43
conductor for which Ohm’s law holds is used, such a situation cannot be realized.
The width, length, and thickness of the transmission line are denoted by w, l, and
d , respectively. Since the magnetic flux density B is common to the two magnetic
materials, the surface current densities are τ 1 = B/μ 1 and τ 2 = B/μ 2 . Hence the
total current is
I =
w
2
(τ 1 + τ 2 ) =
(μ 1 + μ 2 )wB
2μ 1 μ 2
.
(3.27)
Since the magnetic flux is Φ = dlB, the self-inductance, i.e., the magnetic flux stored
by a unit current, is
L =
Φ
I
=
2μ 1 μ 2 dl
(μ 1 + μ 2 )w
.
(3.28)
From the viewpoint of impedance, 1/C corresponds to L. Since corresponds to 1/μ
from (2.2) and (2.22) (note that μ has dimension of (s/m)
2 ), and S and dl are the
areas of the planes normal to E and B, respectively, it can be understood that (3.26)
and (3.28) are analogous to each other. Such effectiveness in education is also one
of the merits of introducing superconductivity.
(4) Derivation of magnetic energy and prediction of the induction law
One of other big merits of introducing superconductivity into electromagnetism is
direct derivation of magnetic energy from mechanical work. The electric energy
can be calculated from the mechanical work needed to carry small electric charges
from infinity until the final desired distribution of electric charge is completed. If
we similarly try to derive the magnetic energy, however, the attractive force works
between currents, and the mechanical work to carry a current is negative. This is
attributed to the electromotive induction. For this reason, it is common that students
learn the magnetic energy, which is derived through equivalent electric energy, after
learning about electromotive induction. That is, learning about dynamic phenomena
is required to learn about the static magnetic phenomena.
On the other hand, it is possible to directly derive the magnetic energy from the
mechanical work done by the Lorentz force in a circuit composed of superconductors.
Suppose that there is a closed circuit made of superconducting slabs, as shown in
Fig. 3.14. The superconducting circuit is composed of a fixed superconducting slab
with three sides and a movable superconducting slab with superconducting contact
between them. For simplicity, b is assumed to be much larger than a. It is assumed
that the magnetic flux is stored in the space surrounded by the circuit. This situation
can be achieved by the following procedure: external magnetic field is first applied
to the superconducting circuit at a temperature higher than the critical temperature,
and then the temperature is lowered below the critical temperature, which makes
the circuit superconducting. Finally, the external magnetic field is removed. This is
called the field-cooled process. In this situation, a shielding current is induced in
the circuit. This process is employed to realize a permanent current. The Lorentz