Question

A charge \( Q \) is distributed over two concentric hollow spheres of radii \( r \) and \( R \) (\( R>r \)) such that their surface charge densities are equal. Find: \begin{enumerate} \item the electric field, and \item the potential at their common center. \end{enumerate}

Hint:

For spherical charge distributions, the electric field outside a shell is as if the charge were concentrated at the center of the sphere. The potential at any point inside a spherical shell is constant and equal to the potential at the surface.

Answer 1

Given: - Two concentric spheres with radii \( r \) and \( R \), and surface charge densities \( \sigma_1 \) and \( \sigma_2 \) such that \( \sigma_1 = \sigma_2 \). - Total charge \( Q \) is distributed between the two spheres. 1. Electric Field: The electric field at a point outside a spherical shell with charge \( Q \) is the same as if all the charge were concentrated at the center of the shell. For a point outside the spheres, the electric field is: \[ E = \frac{1}{4 \pi \epsilon_0} \frac{Q_{\text{total}}}{r^2} \] where \( Q_{\text{total}} = Q \) (the total charge on both spheres). For a point inside the inner sphere (radius \( r \)), the electric field is zero since the enclosed charge is zero. The electric field between the spheres (for \( r<r'<R \)) can be calculated similarly, using the charge enclosed by the Gaussian surface. 2. Potential at the Common Center: The potential at the common center due to a spherical shell of charge is given by the potential due to a point charge: \[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q_{\text{total}}}{R} \] where \( R \) is the radius of the outer sphere and \( Q_{\text{total}} \) is the total charge distributed over the spheres.