Volume 53A, number 4 PHYSICS LETTERS 30 June 1975

T H E T R A N S P O R T O F E L E C T R I C I T Y I N S U P E R C O N D U C T O R S

S U B J E C T T O G R A V I T A T I O N A L O R I N E R T I A L F O R C E S

G. PAPINI Department of Physics and Astronomy, University of Regina,

Regina, Saskatchewan, $4S 0A2, Canada

Received 10 April 1975

In a superconductor subject to gravitational or inertial forces the flow of current is not limited to the penetra- tion depth, but occurs also in the interior.

It is well known that, unlike normal conductors, the flow of steady currents in superconductors is lim- ited to a distance from the surface of the order of the penetration depth ~ [1] (fig. 1). This situation is al- tered, we submit, when the superconductor is subject to gravitational or inertial forces such as those pro- duced in rotation or acceleration. Under the action of these forces the electrons are displaced [2] and the lattice is compressed [3, 4]. The electron displace- ment Ax is obviously very small ~1 as any mismatch between lattice and electrons generate an electric field that opposes the action of the gravitational field.

If the situation of fig. 1 is now modified to include a gravitational field acting along the z-axis, the sta- tionary electrons located in region I are displaced in the direction of Vhoo. It is essential here to realize that as any resulting mismatch between electrons and lattice in I attracts and repels electrons from II and III, respectively, the electric field E produced by hoo is rapidly compensated. The electrons, no longer sup- ported by E against the pull of hoo, are then acceler- ated until the current in the skin and at depths >/?~ equals in value Io the steady current supplied by the source. Typically this situation is reached after a time t o = u/g for a superconductor with incompressible lat- tice if the gravitational field is that of the earth and the source current corresponds to an electron mean velocity u. Since acceleration to velocities in the su-

,1 The value of AX can be estimated from the displacement of the centre of mass of the electron system Ax ~ £/ne = (1/2)(mc2[ne2)~oo, where E, hoo and n represent the gravity induced electric field, the gravitational potential and the electron density, respectively. We adhere in this paper to the notations of ref. [2].

perconductor higher than ]o/ne again tends to gener- ate a mismatch between positive and negative charges, after t o steady state conditions are once more achieved. After t o , therefore, the penetration of the current flow lines in the superconductor appears to be complete 4:2

In general the behaviour of an electron in the su- perconductor after t o can be obtained by means of a coordinate transformation from the original system x' fixed to the lattice to the system x attached to the lattice moving with velocity - u ( x ' ) relative to the electrons. If the lattice is incompressible u is constant. If on the contrary the lattice is compressible n(x') changes according the the equation ~3

2 EF _ c 2 2 47ren(x') = + -~ e--~l l 13 f f V hoo(X )

while

neu (x') =/o

as required by the continuity equation. By applying the transformation

x=x' +u(x')t (1)

to the equation

i ~ t ' (x', t) = ~ o ~'(x ' , t)

,2 For a superconductor with compressible lattice u and n are in general position dependent [3, 4] and the calcula- tion of t o becomes more involved [4].

,3 The symbols are those of ref. [4] eqs. (22), (24), (30) and (31).

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Volume 53A, number 4

I l l [11 n o r m B | o o n d u c t o r

/ I ' I

I I

s u p e r c o n d u c t o r

I

- - " t - - -

;~r

PHYSICS LETTERS 30 June 1975

where [2]

9(.0 ~ ( p e A(x,)_ - , 2 = m c h o ( X )) ,

we obtain after a unitary transformation

• a ~ ( x , t) = ~ ( x , t~ I a ~ Z -

with

= - L ' [vi(u/ t ) ,~l eA + ' 2/1l {P" ~ +- C i--mchoi mui}-"

As expected, the form of 9~ confirms the existence in- side the superconductor of a current neu and of a cor-

responding magnetic field given by curl ~ = (4r:ne/c)u. In addition effects of higher order in u can be predicted. A detailed study of these will be pre- sented elsewhere.

References

[I ] F. London, Superfluids, Vol. 1 (Dover Publ., New York, 1961) p. 37.

[2] S. Dewitt, Phys. Rev. Lett. 16 (1966) 1092: G. Papini, Nuovo Cim. 63B (1969) 549.

[3l T.J. Rieger, Phys. Rev. 2B (1970) 825. [4] M,-C. Leugn, G. Papini and R.G. Rystephanick, Can J.

Phys. 49 (1971) 2754.

n o r m a I c o n d u c t o r

Z

Fig. 1.

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