Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

A conducting metal circular-wire-loop of radius r is placed perpendicular to a
magnetic field which varies with time as

B = B_{0}e$${^{{{ - t} \over r}}}$$ , where B_{0} and $$\tau $$ are constants, at time t = 0. If the resistance of the loop is R then the heat generated in the loop after a long time (t $$ \to $$ $$\infty $$) is :

B = B

A

$${{{\pi ^2}{r^4}B_0^4} \over {2\tau R}}$$

B

$${{{\pi ^2}{r^4}B_0^2} \over {2\tau R}}$$

C

$${{{\pi ^2}{r^4}B_0^2R} \over \tau }$$

D

$${{{\pi ^2}{r^4}B_0^2} \over {\tau R}}$$

Given,

B = B_{0}e$$^{ - {t \over \tau }}$$

Area of the circular loop, A = $$\pi $$ r^{2}

$$ \therefore $$ Flux $$\phi $$ = BA = $$\pi $$ r^{2} B_{0} e$$^{ - {t \over \tau }}$$

Induced emf in the loop,

$$\varepsilon $$ = $$-$$ $${{d\phi } \over {dt}}$$ = $$\pi $$ r^{2}B_{0}$${1 \over \tau }$$e$$^{ - {t \over \tau }}$$

Heat generated

= $$\int\limits_0^ \propto {{i^2}R\,dt} $$

= $$\int\limits_0^ \propto {{{{\varepsilon ^2}} \over R}} \,dt$$

= $${1 \over R}{{{\pi ^2}{r^4}B_0^2} \over {{\tau ^2}}}\int\limits_0^ \propto {{e^{ - {{2t} \over \tau }}}} \,dt$$

= $${{{\pi ^2}{r^4}B_0^2} \over {{\tau ^2}R}} \times {1 \over {\left( { - {2 \over \tau }} \right)}}\left[ {{e^{ - {{2t} \over \tau }}}} \right]_0^ \propto $$

= $${{ - {\pi ^2}{r^4}B_0^2} \over {2{\tau ^2}R}} \times \tau \left( {0 - 1} \right)$$

= $${{{\pi ^2}{r^4}B_0^2} \over {2\tau R}}$$

B = B

Area of the circular loop, A = $$\pi $$ r

$$ \therefore $$ Flux $$\phi $$ = BA = $$\pi $$ r

Induced emf in the loop,

$$\varepsilon $$ = $$-$$ $${{d\phi } \over {dt}}$$ = $$\pi $$ r

Heat generated

= $$\int\limits_0^ \propto {{i^2}R\,dt} $$

= $$\int\limits_0^ \propto {{{{\varepsilon ^2}} \over R}} \,dt$$

= $${1 \over R}{{{\pi ^2}{r^4}B_0^2} \over {{\tau ^2}}}\int\limits_0^ \propto {{e^{ - {{2t} \over \tau }}}} \,dt$$

= $${{{\pi ^2}{r^4}B_0^2} \over {{\tau ^2}R}} \times {1 \over {\left( { - {2 \over \tau }} \right)}}\left[ {{e^{ - {{2t} \over \tau }}}} \right]_0^ \propto $$

= $${{ - {\pi ^2}{r^4}B_0^2} \over {2{\tau ^2}R}} \times \tau \left( {0 - 1} \right)$$

= $${{{\pi ^2}{r^4}B_0^2} \over {2\tau R}}$$

2

Consider an electromagnetic wave propagating in vacuum. Choose the correct
statement :

A

For an electromagnetic wave propagating in +x direction the electric field is $$\vec E = {1 \over {\sqrt 2 }}{E_{yz}}{\mkern 1mu} \left( {x,t} \right)\left( {\hat y - \hat z} \right)$$

and the magnetic field is $$\vec B = {1 \over {\sqrt 2 }}{B_{yz}}{\mkern 1mu} \left( {x,t} \right)\left( {\hat y + \hat z} \right)$$

and the magnetic field is $$\vec B = {1 \over {\sqrt 2 }}{B_{yz}}{\mkern 1mu} \left( {x,t} \right)\left( {\hat y + \hat z} \right)$$

B

For an electromagnetic wave propagating in +x direction the electric field is
$$\vec E = {1 \over {\sqrt 2 }}{E_{yz{\mkern 1mu} }}\left( {y,z,t} \right)\left( {\hat y + \hat z} \right)$$

and the magnetic field is $$\vec B = {1 \over {\sqrt 2 }}{B_{yz{\mkern 1mu} }}\left( {y,z,t} \right)\left( {\hat y + \hat z} \right)$$

and the magnetic field is $$\vec B = {1 \over {\sqrt 2 }}{B_{yz{\mkern 1mu} }}\left( {y,z,t} \right)\left( {\hat y + \hat z} \right)$$

C

For an electromagnetic wave propagating in + y direction the electric field is
$$\overrightarrow E = {1 \over {\sqrt 2 }}{E_{yz{\mkern 1mu} }}\left( {x,t} \right)\widehat y$$

and the magnetic field is $$\vec B = {1 \over {\sqrt 2 }}{B_{yz{\mkern 1mu} }}\left( {x,t} \right)\widehat z$$

and the magnetic field is $$\vec B = {1 \over {\sqrt 2 }}{B_{yz{\mkern 1mu} }}\left( {x,t} \right)\widehat z$$

D

For an electromagnetic wave propagating in + y direction the electric field is
$$\overrightarrow E = {1 \over {\sqrt 2 }}{E_{yz{\mkern 1mu} }}\left( {x,t} \right)\widehat z$$

and the magnetic field is $$\overrightarrow B = {1 \over {\sqrt 2 }}{B_{z{\mkern 1mu} }}\left( {x,t} \right)\widehat y$$

and the magnetic field is $$\overrightarrow B = {1 \over {\sqrt 2 }}{B_{z{\mkern 1mu} }}\left( {x,t} \right)\widehat y$$

As wave is propagating in + x direction, then $$\overrightarrow E $$ and $$\overrightarrow B $$ should be function of $$\left( {x,t} \right)$$ and must be in y $$-$$ z plane.

3

An arc lamp requires a direct current of
10 A at 80 V to function. If it is connected
to a 220 V (rms), 50 Hz AC supply, the
series inductor needed for it to work is
close to :

A

0.044 H

B

0.065 H

C

80 H

D

0.08 H

4

In a coil of resistance 100 $$\Omega $$, a current is induced by changing
the magnetic flux through it as shown in the figure. The
magnitude of change in flux through the coil is:

A

275 Wb

B

200 Wb

C

225 Wb

D

250 Wb

According to Faraday's law of electromagnetic
induction,

$$\varepsilon = {{d\phi } \over {dt}}$$

Also, $$\varepsilon $$ = iR

$$ \therefore $$ $${{d\phi } \over {dt}}$$ = iR

$$ \Rightarrow $$ $$\int {d\phi } = R\int {idt} $$

Magnitude of change in flux (d$$\phi $$) = R × area under current vs time graph

$$ \Rightarrow $$ d$$\phi $$ = $$100 \times {1 \over 2} \times {1 \over 2} \times 10$$ = 250 Wb

$$\varepsilon = {{d\phi } \over {dt}}$$

Also, $$\varepsilon $$ = iR

$$ \therefore $$ $${{d\phi } \over {dt}}$$ = iR

$$ \Rightarrow $$ $$\int {d\phi } = R\int {idt} $$

Magnitude of change in flux (d$$\phi $$) = R × area under current vs time graph

$$ \Rightarrow $$ d$$\phi $$ = $$100 \times {1 \over 2} \times {1 \over 2} \times 10$$ = 250 Wb

Number in Brackets after Paper Name Indicates No of Questions

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Motion *keyboard_arrow_right*

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