question_answer

Two concentric conducting spherical shells of radii \[{{a}_{1}}\]and \[{{a}_{2}}\]\[({{a}_{2}}>{{a}_{1}})\] are charged to potentials \[{{\phi }_{1}}\] and \[{{\phi }_{2}}\], respectively. Find the charge on the inner shell.

A) \[{{q}_{1}}=4\pi {{\varepsilon }_{0}}\left( \frac{{{\phi }_{1}}-{{\phi }_{2}}}{{{a}_{2}}-{{a}_{1}}} \right){{a}_{1}}{{a}_{2}}\]      

B) \[{{q}_{1}}=4\pi {{\varepsilon }_{0}}\left( \frac{{{\phi }_{1}}+{{\phi }_{2}}}{{{a}_{2}}+{{a}_{1}}} \right){{a}_{1}}{{a}_{2}}\]

C) \[{{q}_{1}}=4\pi {{\varepsilon }_{0}}\left( \frac{{{\phi }_{1}}-{{\phi }_{2}}}{{{a}_{2}}+{{a}_{1}}} \right){{a}_{1}}{{a}_{2}}\]

D) \[{{q}_{1}}=4\pi {{\varepsilon }_{0}}\left( \frac{{{\phi }_{1}}+{{\phi }_{2}}}{{{a}_{2}}-{{a}_{1}}} \right){{a}_{1}}{{a}_{2}}\]

Correct Answer: A

Solution :

[a] \[{{\phi }_{1}}=\frac{k{{q}_{1}}}{{{a}_{1}}}+\frac{k{{q}_{2}}}{{{a}_{2}}}\] \[{{\phi }_{2}}=\frac{k{{q}_{1}}}{{{a}_{2}}}+\frac{k{{q}_{2}}}{{{a}_{2}}}\] Solve to get \[{{q}_{1}}=4\pi {{\varepsilon }_{0}}\left[ \frac{{{\phi }_{1}}-{{\phi }_{2}}}{{{a}_{2}}-{{a}_{1}}} \right]{{a}_{1}}{{a}_{2}}\]