Question

Two concentric spheres kept in air have radii \(R\) and \(r\). They have similar charge and equal surface charge density \(\sigma\). The electric potential at their common centre is (\(\epsilon _{0}\) = permittivity of free space)

• A. \(\frac{\sigma (R+r)}{\epsilon _{0}}\)
• B. \(\frac{\sigma (R-r)}{\epsilon _{0}}\)
• C. \(\frac{\sigma (R+r)}{2\epsilon _{0}}\)
• D. \(\frac{\sigma (R+r)}{4\epsilon _{0}}\)

Hint:

The potential at the centre is the sum of potentials due to each sphere, each equal to k Q over its radius.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Potential at the centre of a charged shell is \(V=\dfrac{Q}{4\pi\epsilon_0 R}\). Potentials add as scalars for like charges.

Step 2: Charge on each sphere
\(Q_R=\sigma\,4\pi R^2\) and \(Q_r=\sigma\,4\pi r^2\).

Step 3: Potentials
\(V_R=\dfrac{\sigma4\pi R^2}{4\pi\epsilon_0R}=\dfrac{\sigma R}{\epsilon_0}\), \(V_r=\dfrac{\sigma r}{\epsilon_0}\).

Step 4: Total
\(V=\dfrac{\sigma(R+r)}{\epsilon_0}\), option (A).

Final Answer:
The potential at the centre is sigma times R plus r over epsilon naught. \[ \boxed{\text{(A) }\dfrac{\sigma(R+r)}{\epsilon_{0}}} \]