Question

Consider the force \(F\) on a charge \(q\) due to a uniformly charged spherical shell of radius \(R\) carrying charge \(Q\) distributed uniformly over it. Which one of the following statements is true for \(F\), if \(q\) is placed at distance \(r\) from the centre of the shell?

• A. \(\dfrac{1}{4\pi\varepsilon_0}\dfrac{qQ}{R^2}\gt F\gt0\) for \(r\lt R\)
• B. \(F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{qQ}{r^2}\) for \(r\gt R\)
• C. \(F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{qQ}{r^2}\) for all \(r\)
• D. \(F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{qQ}{R^2}\) for \(r\lt R\)

Hint:

Apply Gauss law with a spherical Gaussian surface: inside the shell the enclosed charge is zero (so \(F=0\)); outside it behaves like a point charge at the centre.

Answer 1

The Correct Option is B

Inside the shell (\(r\lt R\)): a Gaussian sphere of radius \(r\) encloses no charge, so \(E=0\) and \(F=qE=0\). Options (1) (which says \(F\gt0\)) and (4) (non-zero constant) are therefore false.
Outside the shell (\(r\gt R\)): the Gaussian sphere encloses \(Q\), so \(E\cdot4\pi r^2=\dfrac{Q}{\epsilon_0}\Rightarrow E=\dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2}\) and \(F=\dfrac{1}{4\pi\epsilon_0}\dfrac{qQ}{r^2}\). Option (2) is true.
Option (3) is false because the expression is not valid for \(r\lt R\), where \(F=0\).
Answer: (2) \(F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{qQ}{r^2}\) for \(r\gt R\)