and to at the positive surface, and , those on the negative surface
,
(6)
.
(7)
Within the sheet the components vary continuously from and to and .
The equations
,
,
,
(8)
which connect the components , , of the vector-potential due to the current-sheet with the scalar potential , are satisfied if we make
, , .
(9)
We may also obtain these values by direct integration, thus for ,
,
.
Since the integration is to be estimated over the infinite plane sheet, and since the first term vanishes at infinity, the expression is reduced to the second term; and by substituting
for ,
and remembering that depends on and and not on , , , we obtain
,
, by (1).
If is the magnetic potential due to any magnetic or electric system external to the sheet, we may write
,
(10)
and we shall then have
, , ,
(11)
for the components of the vector-potential due to this system.