Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/402

CHAPTER IX.

CONDUCTION THROUGH HETEROGENEOUS MEDIA.

On the Conditions to be Fulfilled at the Surface of Separation between Two Conducting Media.

310.] There are two conditions which the distribution of currents must fulfil in general, the condition that the potential must be continuous, and the condition of 'continuity' of the electric currents.

At the surface of separation between two media the first of these conditions requires that the potentials at two points on opposite sides of the surface, but infinitely near each other, shall be equal. The potentials are here understood to be measured by an electrometer put in connexion with the given point by means of an electrode of a given metal. If the potentials are measured by the method described in Arts. 222, 246, where the electrode terminates in a cavity of the conductor filled with air, then the potentials at contiguous points of different metals measured in this way will differ by a quantity depending on the temperature and on the nature of the two metals.

The other condition at the surface is that the current through any element of the surface is the same when measured in either medium.

Thus, if $V_{1}$ and $V_{2}$ are the potentials in the two media, then at any point in the surface of separation

$V_{1}=V_{2},$

(1)

and if $u_{1},v_{1},w_{1}$ and $u_{2},v_{2},w_{2}$ are the components of currents in the two media, and $l,m,n$ the direction-cosines of the normal to the surface of separation,

$u_{1}l+v_{1}m+w_{1}n=u_{2}l+v_{2}m+w_{2}n.$

(2)

In the most general case the components $u,v,w$ are linear