### JEE Mains Previous Years Questions with Solutions

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1

### AIEEE 2009

A charge $Q$ is placed at each of the opposite corners of a square. A charge $q$ is placed at each of the other two corners. If the net electrical force on $Q$ is zero, then $Q/q$ equals:
A
$-1$
B
$1$
C
$- {1 \over {\sqrt 2 }}$
D
$- 2\sqrt 2$

## Explanation

Let $F$ be the force between $Q$ and $Q.$ The force between $q$ and $Q$ should be attractive for net force on $Q$ to be zero. Let $F'$ be the force between $Q$ and $q.$ For equilibrium

$\sqrt 2 F' = - F$

$\sqrt 2 \times k{{Qq} \over {{\ell ^2}}} = - k{{{Q^2}} \over {{{\left( {\sqrt 2 \ell } \right)}^2}}}$

$\Rightarrow {Q \over q} = - 2\sqrt 2$
2

### AIEEE 2009

(This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.)

Statement-1 : For a charged particle moving from point $P$ to point $Q$, the net work done by an electrostatic field on the particle is independent of the path connecting point $P$ to point $Q.$
Statement-2 : The net work done by a conservative force on an object moving along a closed loop is zero.

A
Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.
B
Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1.
C
Statement- 1 is false, Statement- 2 is true.
D
Statement- 1 true, Statement- 2 is false

## Explanation

Statement $1$ is true.

Statement $2$ is true and is the correct explanation of $(1)$
3

### AIEEE 2009

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

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

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

### AIEEE 2008

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 $(a)$.