### JEE Mains Previous Years Questions with Solutions

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1

### JEE Main 2016 (Offline)

Two identical wires $A$ and $B,$ each of length $'l'$, carry the same current $I$. Wire $A$ is bent into a circle of radius $R$ and wire $B$ is bent to form a square of side $'a'$. If ${B_A}$ and ${B_B}$ are the values of magnetic fields at the centres of the circle and square respectively, then the ratio ${{{B_A}} \over {{B_B}}}$ is:
A
${{{\pi ^2}} \over {16}}$
B
${{{\pi ^2}} \over {8\sqrt 2 }}$
C
${{{\pi ^2}} \over {8}}$
D
${{{\pi ^2}} \over {16\sqrt 2 }}$

## Explanation

Case (a) :

${B_A} = {{{\mu _0}} \over {4\pi }}{I \over R} \times 2\pi$

$= {{{\mu _0}} \over {4\pi }}{I \over {\ell /2\pi }} \times 2\pi$ $\,\,\,\,\,\,\,\,\,$ $\left( {2\pi R = \ell } \right)$

$= {{{\mu _0}} \over {4\pi }}{I \over \ell } \times {\left( {2\pi } \right)^2}$

Case (b) :

${B_B} = 4 \times {{{\mu _0}} \over {4\pi }}{I \over {a/2}}\,\,\,$ $\left[ {\sin \,\,{{45}^ \circ } + \sin \,\,{{45}^ \circ }} \right]$

$= 4 \times {{{\mu _0}} \over {4\pi }} \times {I \over {\ell /8}} \times {2 \over {\sqrt 2 }}$

$= {{{\mu _0}I} \over {4\pi \,\ell }} \times \root {32} \of 2 \,\,\,\,\,\,\left[ {4a = 1} \right]$
2

### JEE Main 2016 (Offline)

A galvanometer having a coil resistance of $100\,\Omega$ gives a full scale deflection, when a currect of $1$ $mA$ is passed through it. The value of the resistance, which can convert this galvanometer into ammeter giving a full scale deflection for a current of $10$ $A,$ is :
A
$0.1\,\Omega$
B
$3\,\Omega$
C
$0.01\,\Omega$
D
$2\,\Omega$

## Explanation

${\rm I}gG = \left( {{\rm I} - {\rm I}g} \right)s$

$\therefore$ ${10^{ - 3}} \times 100 = \left( {10 - {{10}^{ - 3}}} \right) \times S$

$\therefore$ $S \approx 0.01\,\,\Omega$
3

### JEE Main 2015 (Offline)

A rectangular loop of sides $10$ $cm$ and $5$ $cm$ carrying a current $1$ of $12A$ is placed in different orientations as shown in the figures below :

If there is a uniform magnetic field of $0.3$ $T$ in the positive $z$ direction, in which orientations the loop would be in $(i)$ stable equilibrium and $(ii)$ unstable equilibrium ?

A
$(B)$ and $(D)$, respectively
B
$(B)$ and $(C)$, respectively
C
$(A)$ and $(B)$, respectively
D
$(A)$ and $(C)$, respectively

## Explanation

For stable equilibrium $\mathop M\limits^ \to ||\mathop B\limits^ \to$

For unstable equilibrium $\mathop M\limits^ \to ||\left( { - \mathop B\limits^ \to } \right)$
4

### JEE Main 2015 (Offline)

Two long current carrying thin wires, both with current $I,$ are held by insulating threads of length $L$ and are in equilibrium as shown in the figure, with threads making an angle $'\theta '$ with the vertical. If wires have mass $\lambda$ per unit-length then the value of $I$ is :
($g=$ $gravitational$ $acceleration$ )
A
$2\sqrt {{{\pi gL} \over {{\mu _0}}}\tan \theta }$
B
$\sqrt {{{\pi \lambda gL} \over {{\mu _0}}}\tan \theta }$
C
$\sin \theta \sqrt {{{\pi \lambda gL} \over {{\mu _0}\,\cos \theta }}}$
D
$2\sin \theta \sqrt {{{\pi \lambda gL} \over {{\mu _0}\,\cos \theta }}}$

## Explanation

Let us consider $'\ell '$ length of current carrying wire,

At equilibrium

$T\cos \theta = \lambda g\ell$

and $T\sin \theta = {{{\mu _0}} \over {2\pi }}{{I \times Il} \over {2L\sin \theta }}$

$\left[ {\,\,} \right.$ as $\left. {{{{F_B}} \over \ell } = {{{\mu _0}} \over {4\pi }}{{2I \times I} \over {2\ell \sin \theta }}\,\,} \right]$

Therefore, $I = 2\sin \theta \sqrt {{{\pi \lambda gL} \over {{u_0}\cos \theta }}}$