may be applied to current-sheets by substituting for the body supposed to be uniformly magnetized in the direction of with intensity , a current-sheet having the form of its surface, and for which the current-function is
The currents in the sheet will be in planes parallel to that of , and the strength of the current round a slice of thickness will be . The magnetic potential due to this current-sheet at any point outside it will be
At any point inside the sheet it will be
The components of the vector-potential are
These results can be applied to several cases occurring in practice.
675.] (1) A plane electric circuit of any form.
Let be the potential due to a plane sheet of any form of which the surface-density is unity, then, if for this sheet we substitute either a magnetic shell of strength or an electric current of strength round its boundary, the values of and of , , will be those given above.
(2) For a solid sphere of radius ,
when is greater than , | (5) | ||
and | when is less than . | (6) |
Hence, if such a sphere is magnetized parallel to with intensity , the magnetic potential will be
outside the sphere, | (7) | ||
and | inside the sphere. | (8) |
If, instead of being magnetized, the sphere is coiled with wire in equidistant circles, the total strength of current between two small circles whose planes are at unit distance being , then outside the sphere the value of is as before, but within the sphere
This is the case already discussed in Art. 672.