3.3 Merits of Introducing Superconductivity
45
Since (a + x)bh is the volume of the interior space with constant magnetic flux
density, the magnetic energy density is given by
u m =
1
2μ 0
B
2
.
(3.34)
The total current is I = τ h, the self-inductance is
L =
I
=
μ 0 (a + x)b
h
.
(3.35)
Using the self-inductance, the magnetic energy is also expressed as
U m =
1
2
LI
2
=
1
2
I =
1
2L
2
.
(3.36)
The obtained (3.34) is analogous to the facts that the electrostatic energy density
of the space with electric field E is given by
u e =
1
2
0 E
2
,
(3.37)
and (3.36) corresponds also to the electrostatic energy of a capacitor with capacitance
C when electric charge of ±Q is given under electric potential difference V :
U e =
1
2C
Q
2
=
1
2
VQ =
1
2
CV
2
.
(3.38)
As shown in above, isolated current systems are realized for superconducting
electric circuits. In addition, magnetic flux cannot penetrate a superconductor, and
hence, the magnetic flux inside the circuit is preserved. As a result, the electromotive
force does not appear. This is the reason why the magnetic energy can be directly
derived from the magnetic force.
For the usual electric circuits, the magnetic energy cannot be derived from the
magnetic force because of the electromagnetic induction. Hence, it is even possible
to derive the electromagnetic induction law from the difference between a superconducting circuit and a non-superconducting circuit. Here we suppose that current I 1
flows in a straight line and current I 2 flows along a rectangular circuit with two sides
parallel to the straight line, as shown in Fig. 3.15. These are placed on a common
plane. The force on the rectangular circuit is calculated as
F = −
μ 0 I 1 I 2
2π x
b +
μ 0 I 1 I 2
2π (x + a)
b = −
μ 0 abI 1 I 2
2π x(x + a)
.
(3.39)
The force is negative, indicating an attractive force. Here the magnetic energy of this
system is calculated. The self-inductance of the straight line, that of the rectangular
45
Since (a + x)bh is the volume of the interior space with constant magnetic flux
density, the magnetic energy density is given by
u m =
1
2μ 0
B
2
.
(3.34)
The total current is I = τ h, the self-inductance is
L =
I
=
μ 0 (a + x)b
h
.
(3.35)
Using the self-inductance, the magnetic energy is also expressed as
U m =
1
2
LI
2
=
1
2
I =
1
2L
2
.
(3.36)
The obtained (3.34) is analogous to the facts that the electrostatic energy density
of the space with electric field E is given by
u e =
1
2
0 E
2
,
(3.37)
and (3.36) corresponds also to the electrostatic energy of a capacitor with capacitance
C when electric charge of ±Q is given under electric potential difference V :
U e =
1
2C
Q
2
=
1
2
VQ =
1
2
CV
2
.
(3.38)
As shown in above, isolated current systems are realized for superconducting
electric circuits. In addition, magnetic flux cannot penetrate a superconductor, and
hence, the magnetic flux inside the circuit is preserved. As a result, the electromotive
force does not appear. This is the reason why the magnetic energy can be directly
derived from the magnetic force.
For the usual electric circuits, the magnetic energy cannot be derived from the
magnetic force because of the electromagnetic induction. Hence, it is even possible
to derive the electromagnetic induction law from the difference between a superconducting circuit and a non-superconducting circuit. Here we suppose that current I 1
flows in a straight line and current I 2 flows along a rectangular circuit with two sides
parallel to the straight line, as shown in Fig. 3.15. These are placed on a common
plane. The force on the rectangular circuit is calculated as
F = −
μ 0 I 1 I 2
2π x
b +
μ 0 I 1 I 2
2π (x + a)
b = −
μ 0 abI 1 I 2
2π x(x + a)
.
(3.39)
The force is negative, indicating an attractive force. Here the magnetic energy of this
system is calculated. The self-inductance of the straight line, that of the rectangular