JEE Advance - Physics (2019 - Paper 1 Offline - No. 3)
A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V0. A hole with a small area $$\alpha $$4$$\pi $$R2($$\alpha $$ << 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?
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 $${{1 - \alpha } \over {1 - 2\alpha }}$$.
The potential at the center of the shell is reduced by 2$$\alpha $$V0.
The magnitude of electric field at the center of the shell is reduced by $${{\alpha {V_0}} \over {2R}}$$.
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 $${{\alpha {V_0}} \over {2R}}$$.
Explanation
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,
$${V_B} = {{KQ} \over R} - {{K(\alpha Q)} \over {R/2}} = {V_0}(1 - 2\alpha )$$
$$ \therefore $$ $${{{V_C}} \over {{V_B}}} = {{1 - \alpha } \over {1 - 2\alpha }}$$
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