### JEE Mains Previous Years Questions with Solutions

4.5
star star star star star
1

### AIEEE 2006

Two spherical conductors $A$ and $B$ of radii $1$ $mm$ and $2$ $mm$ are separated by a distance of $5$ $cm$ and are uniformly charged. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surfaces of spheres $A$ and $B$ is
A
$4:1$
B
$1:2$
C
$2:1$
D
$1:4$

## Explanation

After connection, ${V_1} = {V_2}$

$\Rightarrow K{{{Q_1}} \over {{r_1}}} = K{{{Q_2}} \over {{r^2}}}$

$\Rightarrow {{Q{}_1} \over {{r_1}}} = {{{Q_2}} \over {{r_2}}}$

The ratio of electric fields

${{{E_1}} \over {{E_2}}} = {{K{{{Q_1}} \over {r_1^2}}} \over {K{{{Q_2}} \over {r_2^2}}}} = {{{Q_1}} \over {r_1^2}} \times {{r_2^2} \over {{Q_2}}}$

$\Rightarrow {{{E_1}} \over {{E_2}}} = {{{r_1} \times r_2^2} \over {r_1^2 \times {r_2}}} \Rightarrow {{{E_1}} \over {{E_2}}} = {{{r_2}} \over {{r_1}}} = {2 \over 1}$

Since the distance between the spheres is large as compared to their diameters, the induced effects may be ignored.
2

### AIEEE 2006

An electric dipole is placed at an angle of ${30^ \circ }$ to a non-uniform electric field. The dipole will experience
A
a translation force only in the direction of the field
B
a translation force only in a direction normal to the direction of the field
C
a torque as well as a translational force
D
a torque only

## Explanation

The electric field will be different at the location of the two charges. Therefore the two forces will be unequal. This will result in a force as well as torque.
3

### AIEEE 2005

A fully charged capacitor has a capacitance $'C'$. It is discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat capacity $'s'$ and mass $'m'.$ If the temperature of the block is raised by $'\Delta T',$ the potential difference $'v'$ across the capacitance is
A
${{mCAT} \over s}$
B
$\sqrt {{{2mCAT} \over s}}$
C
$\sqrt {{{2msAT} \over C}}$
D
${{ms\Delta T} \over C}$

## Explanation

Applying conservation of energy,

${1 \over 2}C{V^2} = m.s\Delta T;\,\,\,$

$V = \sqrt {{{2m.s.\Delta T} \over C}}$
4

### AIEEE 2005

A charged ball $B$ hangs from a silk thread $S,$ which makes angle $\theta$ with a large charged conducting sheet $P,$ as shown in the figure. The surface charge density $\sigma$ of the sheet is proportional to f
A
$\cot \,\theta$
B
$\cos \,\theta$
C
$\tan \,\theta$
D
$\sin \,\theta$

## Explanation

$T\sin \theta = {\sigma \over {{\varepsilon _0}K}}.q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$

$T\cos \theta = mg\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$

Dividing $(i)$ by $(ii),$

$\tan \theta = {{\sigma q} \over {{\varepsilon _0}K.mg}}$
$\therefore$ $\sigma \propto \,\tan \theta$