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1

AIEEE 2009

MCQ (Single Correct Answer)
Let $$P\left( r \right) = {Q \over {\pi {R^4}}}r$$ be the change density distribution for a solid sphere of radius $$R$$ and total charge $$Q$$. For a point $$'p'$$ inside the sphere at distance $${r_1}$$ from the center of the sphere, the magnitude of electric field is :
A
$${Q \over {4\pi \,{ \in _0}\,r_1^2}}$$
B
$${{Qr_1^2} \over {4\pi \,{ \in _0}\,{R^4}}}$$
C
$${{Qr_1^2} \over {3\pi \,{ \in _0}\,{R^4}}}$$
D
$$0$$

Explanation



Let us consider a spherical shell of thickness $$dx$$ and radius $$x.$$ The volume of this spherical shell $$ = 4\pi {r^2}dr.$$

The charge enclosed within shell

$$ = {{{Q_r}} \over {\pi {R^4}}}\left[ {4\pi {r^2}dr} \right]$$

The charge enclosed in a sphere of radius $${r_1}$$ is

$${{4Q} \over {{R^4}}}\int\limits_0^{{r_1}} {{r^3}} dr$$

$$ = {{4Q} \over {{R^4}}}\left[ {{{{r^4}} \over 4}} \right]_0^{{r_1}}$$

$$ = {Q \over {{R^4}}}r_1^4$$

$$\therefore$$ The electric field at point $$p$$ inside the sphere at a distance $${r_1}$$ from the center of the sphere is

$$E = {1 \over {4\pi { \in _0}}}{{\left[ {{Q \over {{R^4}}}r_1^4} \right]} \over {r_1^2}}$$

$$ = {1 \over {4\pi { \in _0}}}{Q \over {{R^4}}}r_1^2$$
2

AIEEE 2008

MCQ (Single Correct Answer)
A thin spherical shell of radius $$R$$ has charge $$Q$$ spread uniformly over its surface. Which of the following graphs most closely represents the electric field $$E(r)$$ produced by the shell in the range $$0 \le r < \infty ,$$ where $$r$$ is the distance from the center of the shell?
A
B
c
C
D

Explanation

The electric field inside a thin spherical shell of radius $$R$$ has charge $$Q$$ spread uniformly over its surface is zero.



Outside the shell the electric field is $$E = k{Q \over {{r^2}}}.$$ These characteristics are represented by graph $$(a)$$.
3

AIEEE 2008

MCQ (Single Correct Answer)
A parallel plate capacitor with air between the plates has capacitance of $$9$$ $$pF.$$ The separation between its plates is $$'d'.$$ The space between the plates has dielectric constant $${k_1}$$ $$=3$$ and thickness $${d \over 3}$$ while the other one has dielectric constant $${k_2} = 6$$ and thickness $${{2d} \over 3}$$. Capacitance of the capacitor is now
A
$$1.8$$ $$pF$$
B
$$45$$ $$pF$$
C
$$40.5$$ $$pF$$
D
$$20.25$$ $$pF$$

Explanation



The given capacitance is equal to two capacitances connected in series where

$${C_1} = {{{k_1}{ \in _0}A} \over {d/3}} = {{3{k_1}{ \in _0}A} \over d}$$

$$ = {{3 \times 3{ \in _0}A} \over d} = {{9{ \in _0}A} \over d}$$

and

$${C_2} = {{{k_2}{ \in _0}A} \over {2d/3}} = {{3{k_2}{ \in _0}A} \over {2d}}$$

$$ = {{3 \times 6{ \in _0}A} \over {2d}} = {{9{ \in _0}A} \over d}$$

The equivalent capacitance $${C_{eq}}$$ is

$${1 \over {C{}_{eq}}} = {1 \over {{C_1}}} + {1 \over {{C_2}}}$$

$$ = {d \over {9{ \in _0}A}} + {d \over {9{ \in _0}A}}$$

$$ = {{2d} \over {9{ \in _0}A}}$$

$$\therefore$$ $${C_{eq}} = {9 \over 2}{{{\varepsilon _0}A} \over d} = {9 \over 2} \times 9pF = 40.5pF$$
4

AIEEE 2007

MCQ (Single Correct Answer)
A parallel plate condenser with a dielectric of dielectric constant $$K$$ between the plates has a capacity $$C$$ and is charged to a potential $$V$$ volt. The dielectric slab is slowly removed from between the plates and then reinserted. The net work done by the system in this process is
A
zero
B
$${1 \over 2}\,\left( {K - 1} \right)\,C{V^2}$$
C
$${{C{V^2}\left( {K - 1} \right)} \over K}$$
D
$$\left( {K - 1} \right)\,C{V^2}$$

Explanation

The potential energy of a charged capacitor before removing the dielectric slat is $$U = {{{Q^2}} \over {2C}}.$$

The potential energy of the capacitor when the dielectric slat is first removed and the reinserted in the gap between the plates is $$U = {{{Q^2}} \over {2C}}$$

There is no change in potential energy, therefore work done is zero.

Questions Asked from Electrostatics

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