Question

Two concentric spheres kept in air have radii 'R' and 'r'. They have similar char equal surface charge density 'σ'. The electric potential at their common centre (E₀ = permittivity of free space)

• A. $\frac{σ(R+r)}{E₀}$
• B. $\frac{σ(R-r)}{E₀}$
• C. $\frac{σ(R+r)}{2E₀}$

Answer 1

Answer: (A)

$\frac{σ(R+r)}{E₀}$

Answer 2

For a spherical shell of radius R with uniform surface charge density σ, the potential (inside and on the surface) is given by

$V = \frac{Q}{4\pi \epsilon_0 R}$

where the charge $Q = \sigma \times 4\pi R^2$. Thus,

$V = \frac{\sigma \,4\pi R^2}{4\pi \epsilon_0 R}=\frac{\sigma R}{\epsilon_0}$.

Similarly, for the inner sphere of radius r,

$V = \frac{\sigma r}{\epsilon_0}$.

Since the spheres are concentric, the potential at the centre is just the algebraic sum:

$V_{\text{centre}} = \frac{\sigma R}{\epsilon_0} + \frac{\sigma r}{\epsilon_0} = \frac{\sigma (R+r)}{\epsilon_0}$.