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

Considered Systems: A conducting spherical shell of radius \(R\) containing a concentric conducting ball of radius \(R/10\).

System	Shell Charge	Inner Ball Charge	Net Enclosed Charge \(Q_{\text{net}}\)
A	\(+6q\)	\(-2q\)	\(+4q\)
B	\(-4q\)	\(+8q\)	\(+4q\)
C	\(14q\)	\(-10q\)	\(+4q\)

Core Principle (Gauss’s Law, Spherical Symmetry)

For radial distances \(r \ge R\), the electric field magnitude is determined solely by the total enclosed charge: \[ |\mathbf{E}(r)|=\frac{1}{4\pi\varepsilon_0}\frac{|Q_{\text{net}}|}{r^2}. \] As all systems exhibit \(Q_{\text{net}}=+4q\), at a distance of \(r=3R\), the field is: \[ |\mathbf{E}(3R)|=\frac{1}{4\pi\varepsilon_0}\frac{4q}{(3R)^2}=\frac{4kq}{9R^2}. \]

Comparison of Results:
The magnitudes of the electric fields at \(3R\) for systems A, B, and C are in the ratio \(1:1:1\). The magnitude for each is \(|\mathbf{E}(3R)|=\frac{4kq}{9R^2}\).

Clarification: The distance \(3R\) is used for comparison, not the final answer. The fields are equal because only the net enclosed charge is relevant outside the system, and all three systems share the same net charge of \(+4q\).