# Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/407

314.]

MEDIUM CONTAINING SMALL SPHERES.

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314.] If there are ${\displaystyle n}$ spheres of radius ${\displaystyle a_{1}}$ and resistance ${\displaystyle k_{1},}$ placed in a medium whose resistance is ${\displaystyle k_{2},}$ at such distances from each other that their effects in disturbing the course of the current may be taken as independent of each other, then if these spheres are all contained within a sphere of radius ${\displaystyle a_{2},}$ the potential at a great distance from the centre of this sphere will be of the form

 ${\displaystyle V=\left(A+nB{\frac {1}{r^{2}}}\right)\cos \theta ,}$ (12)

where the value of ${\displaystyle B}$ is

 ${\displaystyle B={\frac {k_{1}-k_{2}}{2k_{1}+k_{2}}}a_{1}^{3}A.}$ (13)

The ratio of the volume of the ${\displaystyle n}$ small spheres to that of the sphere which contains them is

 ${\displaystyle p={\frac {na_{1}^{3}}{a_{2}^{3}}}.}$ (14)

The value of the potential at a great distance from the sphere may therefore be written

 ${\displaystyle V=\left(A+pa_{2}^{3}{\frac {k_{1}-k_{2}}{2k_{1}+k_{2}}}{\frac {1}{r^{2}}}\right)\cos \theta .}$ (15)

Now if the whole sphere of radius ${\displaystyle a_{2}}$ had been made of a material of specific resistance ${\displaystyle K,}$ we should have had

 ${\displaystyle V=\left\{A+a_{2}^{3}{\frac {K-k_{2}}{2K+k_{2}}}{\frac {1}{r^{2}}}\right\}\cos \theta .}$ (16)

That the one expression should be equivalent to the other,

 ${\displaystyle K={\frac {2k_{1}+k_{2}+p(k_{1}-k_{2})}{2k_{1}+k_{2}-2p(k_{1}-k_{2})}}k_{2}.}$ (17)

This, therefore, is the specific resistance of a compound medium consisting of a substance of specific resistance ${\displaystyle k_{2},}$ in which are disseminated small spheres of specific resistance ${\displaystyle k_{1},}$ the ratio of the volume of all the small spheres to that of the whole being ${\displaystyle p.}$ In order that the action of these spheres may not produce effects depending on their interference, their radii must be small compared with their distances, and therefore ${\displaystyle p}$ must be a small fraction.

This result may be obtained in other ways, but that here given involves only the repetition of the result already obtained for a single sphere.

When the distance between the spheres is not great compared with their radii, and when ${\displaystyle {\frac {k_{1}-k_{2}}{2k_{1}+k_{2}}}}$ is considerable, then other terms enter into the result, which we shall not now consider. In consequence of these terms certain systems of arrangement of