1

### JEE Main 2017 (Online) 8th April Morning Slot

A small circular loop of wire of radius a is located at the centre of a much larger circular wire loop of radius b. The two loops are in the same plane. The outer loop of radius b carries an alternating current I = Io cos ($\omega$t). The emf induced in the smaller inner loop is nearly :
A
${{\pi {\mu _o}{I_o}} \over 2}.{{{a^2}} \over b}\,\omega \sin \left( {\omega t} \right)$
B
${{\pi {\mu _o}{I_o}} \over 2}.{{{a^2}} \over b}\,\omega \cos \left( {\omega t} \right)$
C
$\pi {\mu _o}{I_o}\,{{{a^2}} \over b}\omega \sin \left( {\omega t} \right)$
D
${{\pi {\mu _o}{I_o}\,{b^2}} \over a}\omega \cos \left( {\omega t} \right)$

## Explanation

Mutual inductance,

M = ${{{\mu _0}\pi {N_1}{N_2}\,{a^2}} \over {2b}}$

here ${{N_1}}$ = N2 = 1

$\therefore\,\,\,$ M = ${{{\mu _0}\pi {a^2}} \over {2b}}$

Current I = I0 cos ($\omega$t)

e = $-$ M ${{dI} \over {dt}}$

= $-$ ${{{\mu _0}\pi {a^2}} \over {2b}}$ ${d \over {dt}}$ (I0 cos $\omega$t)

= + ${{{\mu _0}\pi {a^2}} \over {2b}}$ I0 $\omega$ sin $\omega$t

= ${{\pi {\mu _0}{I_0}} \over 2}$ . ${{{a^2}} \over b}$ $\omega$ sin $\omega$ t
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### JEE Main 2017 (Online) 9th April Morning Slot

A sinusoidal voltage of peak value 283 V and angular frequency 320/s is applied to a series LCR circuit. Given that R=5 $\Omega$, L=25 mH and C=1000 $\mu$F. The total impedance, and phase difference between the voltage across the source and the current will respectively be :
A
10 $\Omega$ and tan$-$1 $\left( {{5 \over 3}} \right)$
B
$7\,\Omega$ and 45o
C
$10\,\Omega$ and tan$-$1$\left( {{8 \over 3}} \right)$
D
$7\,\Omega$ and tan$-$1$\left( {{5 \over 3}} \right)$
3

### JEE Main 2017 (Online) 9th April Morning Slot

The electric field component of a monochromatic radiation is given by

$\overrightarrow E$ = 2 E0 $\widehat i$ cos kz cos $\omega$t

Its magnetic field $\overrightarrow B$ is then given by :
A
${{2{E_0}} \over c}$ $\widehat j$ sin kz cos $\omega$t
B
$-$ ${{2{E_0}} \over c}$ $\widehat j$ sin kz sin $\omega$t
C
${{2{E_0}} \over c}$ $\widehat j$ sin kz sin $\omega$t
D
${{2{E_0}} \over c}$ $\widehat j$ cos kz cos $\omega$t
4

### JEE Main 2018 (Offline)

In an a.c. circuit, the instantaneous e.m.f. and current are given by
e = 100 sin 30 t
i = 20 sin $\left( {30t - {\pi \over 4}} \right)$
In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively
A
50, 0
B
50, 10
C
${{1000} \over {\sqrt 2 }},10$
D
${{50} \over {\sqrt 2 }}$

## Explanation

Wattless current,

here  $\phi$  is the angle between i and e.

Average power,

Pav = Vrms Irms cos$\phi$

= ${{100} \over {\sqrt 2 }} \times {{20} \over {\sqrt 2 }}$ cos${\pi \over 4}$

= ${{1000} \over {\sqrt 2 }}$ watt.