364
CONDUCTION IN HETEROGENEOUS MEDIA.
[313.
then
|
| (7) |
The difference between
the undisturbed coefficient, and
its value in the hollow within the spherical shell, is
|
![{\displaystyle A_{3}-A_{1}=(k_{2}-k_{1})^{2}i(i+1)\left(1-({\frac {a_{1}}{a_{2}}})^{2i+1}\right)CA_{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aca576593e337f6c1cae49ba1539137fe24642d) | (8) |
Since this quantity is always positive whatever be the values of
and
it follows that, whether the spherical shell conducts better or worse than the rest of the medium, the electrical action within the shell is less than it would otherwise be. If the shell is a better conductor than the rest of the medium it tends to equalize the potential all round the inner sphere. If it is a worse conductor, it tends to prevent the electrical currents from reaching the inner sphere at all.
The case of a solid sphere may be deduced from this by making
or it may be worked out independently.
313.] The most important term in the harmonic expansion is that in which
for which
|
| (9) |
The case of a solid sphere of resistance
may be deduced from this by making
We then have
|
| (10) |
It is easy to shew from the general expressions that the value of
in the case of a hollow sphere having a nucleus of resistance
surrounded by a shell of resistance
is the same as that of a uniform solid sphere of the radius of the outer surface, and of resistance
, where
|
| (11) |