Question

Consider three metal spherical shells A, B, and C, each of radius \( R \). Each shell has a concentric metal ball of radius \( R/10 \). The spherical shells A, B, and C are given charges \( +6q, -4q, \) and \( 14q \) respectively. Their inner metal balls are also given charges \( -2q, +8q \) and \( -10q \) respectively. Compare the magnitude of the electric fields due to shells A, B, and C at a distance \( 3R \) from their centers.

Hint:

The electric field outside a spherical shell depends on the total charge on the shell. It is treated as if the charge is concentrated at the center of the shell.

Answer 1

For a spherical shell, the electric field outside the shell at a distance \( r \) from the center is given by the formula: \[ E = \frac{kQ}{r^2} \] where \( Q \) is the total charge on the shell, and \( k \) is Coulomb's constant. At a distance \( 3R \) from the center, the electric field due to each shell is the same as if the entire charge were concentrated at the center of the shell. Therefore, the electric fields due to shells A, B, and C at a distance \( 3R \) are given by: \[ E_A = \frac{k(6q)}{(3R)^2} = \frac{k(6q)}{9R^2} \] \[ E_B = \frac{k(-4q)}{(3R)^2} = \frac{k(-4q)}{9R^2} \] \[ E_C = \frac{k(14q)}{(3R)^2} = \frac{k(14q)}{9R^2} \] Thus, the magnitude of the electric fields is: \[ |E_A| = \frac{6kq}{9R^2}, \quad |E_B| = \frac{4kq}{9R^2}, \quad |E_C| = \frac{14kq}{9R^2} \] The electric field due to shell C is the largest, followed by shell A, and then shell B.