44
3 Effects of the Introduction of Superconductivity into Electromagnetism
Fig. 3.14 Magnetic flux
stored in closed circuit made
of superconducting slabs
force is exerted on the movable superconducting slab that carries the current. Even
if the distance between the slabs changes from a to a + x, the magnetic flux inside
the circuit is preserved. This is because the magnetic flux cannot pass through the
superconducting region to go outside. Hence, the magnetic flux density changes to
B =
(a + x)b
(3.29)
and the density of the current flowing on the inner surface of the superconductors
changes to
τ =
B
μ 0
=
μ 0 (a + x)b
.
(3.30)
Since the magnetic flux density produced by the fixed superconducting slab is half
of the value given by (3.29), the force exerted on the movable slab is
F =
1
2
τ Bbh =
2 h
2μ 0 (a + x)
2 b
.
(3.31)
Since this circuit is isolated from its surroundings, this force originates from the
variation in the magnetic energy U m of this circuit. From the relationship,
F = −
∂U m
∂x
,
(3.32)
the magnetic energy is obtained as
U m =
2 h
2μ 0 (a + x)b
=
1
2μ 0
B
2
(a + x)bh.
(3.33)