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12-9 Two large parallel plane conducting plates are a distance d apart. The region between them...

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Physics

12-9 Two large parallel plane conducting plates are a distance dapart. The region between them is filled with two l.i.h. materialswhose surface of separation is a plane parallel to the plates. Thefirst material (with properties σ1 and Ɛ1) is of thickness x, whilethe second material (σ2, Ɛ2) has thickness d - x. There is a steadycurrent between the plates that are kept at constant potentials ofφ1 and φ2. Find the potential at the surface of separation of thetwo media, and the free surface charge density there.

The answers are:
Potential: [σ1(d - x)φ1 + σ2(x)φ2]/[σ1(d - x) +σ2(x)];

Free Surface Charge Density: (σ1*Ɛ2 - σ2*Ɛ1)(φ1 - φ2)/[σ1(d - x)+σ2(x)]

Please show all work.

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