Question 796: Electrostatics & Capacitance

Question

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V
0
. A hole with a small area $$\alpha $$4$$\pi $$R
2
($$\alpha $$ << 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Accepted Answer

Given, V at surface of the sphere
$${V_0} = {{KQ} \over R}$$
Here, $$K = {1 \over {4\pi {\varepsilon _0}}}$$ = constant
V at point C,
$${V_C} = {{KQ} \over R} - {{K\alpha Q} \over R} = {V_0}(1 - \alpha )$$
V at point B,
$${V_B} = {{KQ} \over R} - {{K(\alpha Q)} \over {R/2}} = {V_0}(1 - 2\alpha )$$
$$ 	herefore $$ $${{{V_C}} \over {{V_B}}} = {{1 - \alpha } \over {1 - 2\alpha }}$$

Answer

The ratio of the potential at the center of the shell of that of the point at $${1 \over 2}$$R from center towards the hole will be $${{1 - \alpha } \over {1 - 2\alpha }}$$.

Answer

The potential at the center of the shell is reduced by 2$$\alpha $$V
0
.

Answer

The magnitude of electric field at the center of the shell is reduced by $${{\alpha {V_0}} \over {2R}}$$.

Answer

The magnitude of electric field at a point, located on a line passing through the hole and shell's center, on a distance 2R from the center of the spherical shell will be reduced by $${{\alpha {V_0}} \over {2R}}$$.

Electrostatics & Capacitance

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