Why is the induced charge on inner shell of a spherical capacitor considered? [closed]

In the above type of spherical capacitor, it was given thas as the bettery is only connected to outer shell charge will only be present there and as the other end is connected to earth(thereby assume that the other plate is at infinity).

But my question is wouldn't there be an induced charge on the inner shell which would produce some capacitance between them?

• Charge is conserved, and if the inner sphere is insulated... induction won't charge it. Commented Mar 3, 2021 at 3:38
• But it isn't specified if the inner shell is insulated it is a hollow conducting sphere? Commented Mar 3, 2021 at 3:49

1 Answer

The inner conducting spherical shell is uncharged (not connected to any source that can impart it a charge), and, thus, at zero potential. Note that the $$E$$ field inside a conductor must be zero.

By Gauss's law, we have that electric flux is through a surface is proportional to the charge enclosed by the surface. Due to symmetry, you can note that any possible electric field in this system must be radial.

Consider a Gaussian surface coinciding with the inner spherical shell. As the electric field must be zero inside the conductor, and we know any field has to be radially symmetric, we claim that flux through this surface is 0.

As there is no charge enclosed by the inner surface of the inner shell, we claim that there is no surface charge on the inner surface. As the inner conductor is neutral, there is no charge buildup anywhere on the inner conductor.

No induction of charge means that the two shells are not coupled by an electric field, and so capacitance is absent.