Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/405

In this way we can find the conditions which each term of the harmonic expansion of the potential must satisfy for any number of strata bounded by concentric spherical surfaces.

312.] Let us suppose the radius of the first spherical surface to be $a_{1},$ and let there be a second spherical surface of radius $a_{2}$ greater than $a_{1}$ beyond which the specific resistance is $k_{3}.$ If there are no sources or sinks of electricity within these spheres there will be no infinite values of $V,$ and we shall have $B_{1}=0.$

We then find for $A_{3}$ and $B_{3},$ the coefficients for the outer medium,

$\left.{\begin{aligned}A_{3}k_{1}k_{2}(2i+1)^{2}=&{\bigg [}\{k_{1}(i+1)+k_{2}i\}\{k_{2}(i+1)+k_{3}i\}\\&\;\;+i(i+1)(k_{1}-k_{2})(k_{2}-k_{3})\left({\frac {a_{1}}{a_{2}}}\right)^{2i+1}{\bigg ]}A_{1},\\B_{3}k_{1}k_{2}(2i+1)^{2}=&[i\{k_{1}(i+1)+k_{2}i\}(k_{2}-k_{3})a_{2}^{2i+1}\\&\;\;+i(k_{1}-k_{2})\{k_{2}i+k_{3}(i+1)\}a_{1}^{2i+1}]A_{1}.\end{aligned}}\right\}$

(6)

The value of the potential in the outer medium depends partly on the external sources of electricity, which produce currents independently of the existence of the sphere of heterogeneous matter within, and partly on the disturbance caused by the introduction of the heterogeneous sphere.

The first part must depend on solid harmonics of positive degrees only, because it cannot have infinite values within the sphere. The second part must depend on harmonics of negative degrees, because it must vanish at an infinite distance from the centre of the sphere.

Hence the potential due to the external electromotive forces must be expanded in a series of solid harmonics of positive degree. Let $A_{3}$ be the coefficient of one these, of the form

$A_{3}S_{i}r^{i}$

Then we can find $A_{1},$ the corresponding coefficient for the inner sphere by equation (6), and from this deduce $A_{2},B_{2}$ and $B_{3}.$ Of these $B_{3}$ represents the effect on the potential in the outer medium due to the introduction of the heterogeneous spheres.

Let us now suppose $k_{3}=k_{1},$ so that the case is that of a hollow shell for which $k=k_{2},$ separating an inner from an outer portion of the same medium for which $k=k_{1}.$

If we put

$C={\frac {1}{(2i+1)^{2}k_{1}k_{2}+i(i+1)(k_{2}-k_{1})^{2}\left(1-({\frac {a_{1}}{a_{2}}})^{2i+1}\right)}},$