Page:Elementary Text-book of Physics (Anthony, 1897).djvu/357

§ 290] shells, and likewise a current in a circuit coincident with the boundary of the finite shell may be built up of the elementary circuits corresponding to the elementary shells; for the current in each of the elementary circuits will be everywhere neutralized by the equal and opposite currents of the contiguous circuits except at the boundary of the surface occupied by the circuits. At the boundary the currents of the elementary circuits are in the same direction, and are not neutralized by other currents; they are therefore equivalent to the current in the circuit coincident with the boundary of the shell. This reasoning is plain from Fig. 84. If the strength of a finite shell be constant, the potential of the shell is $$j \Omega,$$ where $$\Omega$$ is the solid angle subtended by the shell from an external point. The potential of the equivalent current is therefore $$i \Omega.$$

290. Multiply-valued Potential of the Current.—There is an important difference between the potential due to a current and that due to its equivalent magnetic shell, owing to the fact that the substance of the shell interrupts the field so that the potential within it does not follow the same law as that outside it. If we suppose that the shell is plane, the potential at a point on its positive face is $$2\pi j$$ and that at the corresponding point on the negative face is $$-2 \pi j,$$ so that the work done in transferring a unit pole from one point to the other is $$4 \pi j$$ (§ 243). The pole can only be brought back to the point from which it started by carrying it around the edge of the shell and by doing upon it the work $$4 \pi j,$$ so that when it is returned to the starting-point the work done upon it is zero. If, on the other hand, the pole is moving under the influence of the circuit equivalent to the magnetic shell, the work done in transferring it outside the circuit from the positive to the negative face is equal to $$4 \pi i.$$ But it is not necessary to carry it again outside the circuit to return it to the starting-point. This may be accomplished by, an infinitesimal displacement through the plane of the circuit; and since the force is everywhere finite, no