A charge Q is spread non-uniformly on the surface of a hollow sphere of radius R, such that the charge density is given by σ=σ0(1 sin θ) where θ is the usual polar angle. The potential at the centre of the sphere is

Question

A charge Q is spread non-uniformly on the surface of a hollow sphere of radius R, such that the charge density is given by σ=σ0(1 sin θ) where θ is the usual polar angle. The potential at the centre of the sphere is (A) (Q/2πεoR) (B) (Q/πεoR) (C) (Q/8πεoR) (D) (Q/4πεoR)

Tardigrade Admin · Accepted Answer

Potential due to an elemental charge dQ at centre of sphere is dV=(kdQ/R) Potential due to complete charge distribution at centre is V= oint dV= oint(kdQ/R) =(k/R) oint dQ=(kQ/R) or V=(kQ/R)=(Q/4πεoR)

Q. A charge $Q$ is spread non-uniformly on the surface of a hollow sphere of radius $R$ , such that the charge density is given by $σ = σ_{0} (1 sin θ)$ where $θ$ is the usual polar angle. The potential at the centre of the sphere is

$\frac{Q}{2 π ε _{o} R}$

$\frac{Q}{π ε _{o} R}$

$\frac{Q}{8 π ε _{o} R}$

$\frac{Q}{4 π ε _{o} R}$

Potential due to an elemental charge $d Q$ at centre of sphere is
$d V = \frac{k d Q}{R}$
Potential due to complete charge distribution at centre is
$V = \oint d V = \oint \frac{k d Q}{R}$
$= \frac{k}{R} \oint d Q = \frac{k Q}{R}$
or $V = \frac{k Q}{R} = \frac{Q}{4 π ε _{o} R}$

Q. A charge Q is spread non-uniformly on the surface of a hollow sphere of radius R, such that the charge density is given by σ=σ0​(1sinθ) where θ is the usual polar angle. The potential at the centre of the sphere is

2πεo​RQ​

πεo​RQ​

8πεo​RQ​

4πεo​RQ​