A spherical electrode of radius a is surrounded by a concentric spherical shell

Question

A spherical electrode of radius a is surrounded by a concentric spherical shell of radius b. The space between
the two conductors is filled with material whose conductivity varies with distance from the common center of the
spheres (=K/r, where K is a constant). If the potential difference of V volts is maintained between the spheres,
with the inner electrode grounded, what is the electric potential at a distance r (a<r<b) from the center?

To solve this question you need to know steady electric currents and static electric field, both are sub-subjects of electromagnetism.

Explanation / Answer

In this case as we are having two spherical shells with radius a and radius b and the potential is given as V.

Now let us suppose that the charge on the smaller shell is q and bigger shell is q'

as we need to calculate the potential at a<r<b, for thsi we consider a point at x distance given as r.

So the potential at this point is wil be:

as r is outer to the smaller sphere, so due to the q, potential will be given as:-

V= kq/x (x is the distance of point between the two spheres), q is the charge, k is the constant

and due to q' i.e. outer sphere will be:-

radius this point is b, as it is euipotential surface so the potential at each point will be same. sot he potential will be V= kq'/b

and the total potential will be :-

V = kq/x + kq'/b

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