Electrostatic Potential Energy stored outside a shell.

[cheapstrike]

Homework Statement

A point charge +Q is placed at the centre of an isolated conducting shell of radius R. Find the electrostatic potential energy stored outside the spherical shell if the shell also contains a charge +Q distributed uniformly over it.

Homework Equations

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E=kQ/r².

dU/dV=(1/2)ε_oE², where V is volume, U is potential energy.

The Attempt at a Solution

The charge +Q inside the conductor will induce -Q charge on the inside surface, which further leads to +2Q charge on the outer surface of the shell. Therefore E=2kQ/r².

My question is, why have we used the formula dU/dV=(1/2)ε_oE². How did we derive it? Isn't it equal to the energy of a charged capacitor.

Sorry if this a bad question, I just want to know how did we derive this formula.

Thanks.

 

[kuruman]

My question is, why have we used the formula dU/dV=(1/2)εoE2

For the same reason that we use mass density ρ to find the mass of an extended object by integration.

How did we derive it?

That you can find in any intermediate-level book on Electricity and Magnetism.

Isn't it equal to the energy of a charged capacitor.

No, it is not. It is an energy density, i.e. energy per unit volume (Joules/m³ in SI units). To do this problem you need to use Gauss's law to find the electric field outside the shell, find the energy density and integrate that over all space outside the shell.