Surface Charge on Conducting Sphere

Electricity and Magnetism Level 4

A conducting sphere of radius \(R\) with a layer of charge \(Q\) distributed on its surface has the electric potential everywhere in space:

\[V = \begin{cases} \dfrac{1}{4\pi \epsilon_0} \dfrac{QR^2}{r^3} \sin \theta \cos \theta \cos \phi, \qquad &r>R \\ \dfrac{1}{4\pi \epsilon_0} \dfrac{Qr^2}{R^3} \sin \theta \cos \theta \cos \phi, \qquad &r<R \end{cases}.\]

Which of the following gives the surface charge density on the surface of the sphere?

Note: recall that the change in electric field across either side of a conductor is equal to \(\dfrac{\sigma}{\epsilon_0}\) where \(\sigma\) is the surface charge density.


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