Joint Entrance Examination

Graduate Aptitude Test in Engineering

Strength of Materials Or Solid Mechanics

Structural Analysis

Construction Material and Management

Reinforced Cement Concrete

Steel Structures

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

Hydrology

Irrigation

Geomatics Engineering Or Surveying

Environmental Engineering

Transportation Engineering

Engineering Mathematics

General Aptitude

1

A series *LR* circuit is connected to a voltage source with

V(t) = V_{0 } sin$$\Omega $$t. After very large time, current *I(t)* behaves as

(t_{0} >> $${L \over R}$$) :

V(t) = V

(t

A

B

C

D

Current in LR circuit,

$${\rm I} = {{{V_0}} \over {\sqrt {{R^2} + {w^2}{L^2}} }}\sin \left( {\omega t - {\pi \over 2}} \right)$$

it will be a sinusoidal wave.

$${\rm I} = {{{V_0}} \over {\sqrt {{R^2} + {w^2}{L^2}} }}\sin \left( {\omega t - {\pi \over 2}} \right)$$

it will be a sinusoidal wave.

2

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

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

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

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

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

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

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

3

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.

4

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

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