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:

According to Gauss’s law, the electric field outside a spherical shell behaves as if all the charge were concentrated at its center.

Answer 1

Given (each system considered separately): A conducting spherical shell of radius \(R\) with a concentric conducting ball of radius \(R/10\).

System	Shell charge	Inner ball charge	Net enclosed charge \(Q_{\text{net}}\)
A	\(+6q\)	\(-2q\)	\(6q-2q=+4q\)
B	\(-4q\)	\(+8q\)	\(-4q+8q=+4q\)
C	\(14q\)	\(-10q\)	\(14q-10q=+4q\)

Key idea (Gauss’s law, spherical symmetry)

For \(r \ge R\), the field depends only on the total enclosed charge: \[ |\mathbf{E}(r)|=\frac{1}{4\pi\varepsilon_0}\frac{|Q_{\text{net}}|}{r^2}. \] Since each system has \(Q_{\text{net}}=+4q\), at \(r=3R\): \[ |\mathbf{E}(3R)|=\frac{1}{4\pi\varepsilon_0}\frac{4q}{(3R)^2}=\frac{4kq}{9R^2}. \]

Result (comparison):
\[ |\mathbf{E}_A|:|\mathbf{E}_B|:|\mathbf{E}_C| = 1:1:1, \quad \text{with} \quad |\mathbf{E}(3R)|=\frac{4kq}{9R^2}\ \text{for each}. \]

Note: “\(3R\)” is the distance at which we compare the fields, not the answer. The equality holds because only the net enclosed charge matters outside, and all three systems have the same net \(+4q\).

Answer 2

The electric field at a distance \( 3R \) from the center of a spherical shell depends only on the net charge enclosed and is given by Gauss’s law: \[ E = \frac{1}{4\pi \epsilon_0} \frac{Q_{\text{net}}}{r^2} \] where \( Q_{\text{net}} \) is the total charge enclosed by each shell. 

 Step 1: Calculate Net Charge on Each Shell - For Shell A: \[ Q_A = 6q + (-2q) = 4q \] - For Shell B: \[ Q_B = -4q + 8q = 4q \] - For Shell C: \[ Q_C = 14q + (-10q) = 4q \] Since the total charge enclosed for all three shells is the same (\( 4q \)), the magnitude of the electric field at a distance \( 3R \) is identical for all: \[ E_A = E_B = E_C = \frac{1}{4\pi \epsilon_0} \frac{4q}{(3R)^2} \] Thus, the electric fields due to shells A, B, and C at a distance \( 3R \) are equal.