1
GATE ECE 2014 Set 1
Numerical
+2
-0
In spherical coordinates, let $${{{\widehat a}_{_0}},\,{{\widehat a}_{_\phi }}}$$ denote unit vectors along the $$\theta ,\,\,\phi$$directions. $$E = {{100} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\theta }}\,\,V/m$$$and $$H = {{0.265} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\phi }}\,\,A/m$$$

Represent the electric and magnetic field components of the EM wave at large distances $$r$$ from a dipole antenna, in free space. The average power $$(W)$$ crossing the hemispherical shell located at $$r = 1\,\,km$$, $$0 \le \theta \le \pi /2$$ _______.

2
GATE ECE 2013
+2
-0.6
A monochromatic plane wave of wavelength $$\lambda = 600$$ is propagating in the direction as shown in the figure below. $${\overrightarrow E _i},\,{\overrightarrow E _r}$$ and $${\overrightarrow E _t}$$ denote incident, reflected, and transmitted electric field vectors associated with the wave.

The expression for $${\overrightarrow E _r}$$ is

A
$$0.23{{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}\left( {x - z} \right)} \over {3\sqrt 2 }}}}\,\,\,V/m$$
B
$$- {{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}z} \over 3}}}\,\,\,V/m$$
C
$$0.44{{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}\left( {x - z} \right)} \over {3\sqrt 2 }}}}\,\,\,V/m$$
D
$${{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}\left( {x + z} \right)} \over 3}}}\,\,V/m$$
3
GATE ECE 2013
+2
-0.6
A monochromatic plane wave of wavelength $$\lambda = 600$$ is propagating in the direction as shown in the figure below. $${\overrightarrow E _i},\,{\overrightarrow E _r}$$ and $${\overrightarrow E _t}$$ denote incident, reflected, and transmitted electric field vectors associated with the wave.

The angle of incidence $${\theta _i}$$ and the expression for $${\overrightarrow E _i}$$ are

A
$${60^ \circ }\,\,\,and\,\,{{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}\left( {x + z} \right)} \over {3\sqrt 2 }}}}\,\,V/m$$
B
$${45^ \circ }\,\,\,and\,\,{{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}z} \over 3}}}\,\,V/m$$
C
$${45^ \circ }\,\,\,and\,\,{{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}\left( {x + z} \right)} \over {3\sqrt 2 }}}}\,\,V/m$$
D
$${60^ \circ }\,\,\,and\,\,{{{E_0}} \over {\sqrt 2 }}\left( {{{\widehat a}_x} - {{\widehat a}_z}} \right){e^{ - j{{\pi \times {{10}^4}z} \over 3}}}\,\,V/m$$
4
GATE ECE 2011
+2
-0.6
The electric and magnetic fields for a TEM wave of frequency $$14 GHz$$ in a homogeneous medium of relative permittivity $${\varepsilon _r}$$ and relative permeability $${\mu _r} = 1$$ are given by $$\overrightarrow E = {E_p}\,\,{e^{j\left( {\omega t - 280\pi y} \right)}}\,\,{\widehat u_z}\,\,V/m$$$$$\overrightarrow H = \,\,3\,\,{e^{j\left( {\omega \,t - 280\,\,\pi \,y} \right)}}\,\,\widehat u{\,_x}\,\,A/m$$$

Assuming the speed of light in free space to be $$3\,\, \times {10^8}\,\,\,m/s,$$ the intrinsic impedance of free space to be $$120\,\,\,\pi$$, the relative permittivity $${\varepsilon _r}$$ of the medium and the electric field amplitude $${E_p}$$ are

A
$${\varepsilon _r} = 3,\,\,{E_p} = 120\,\pi$$
B
$${\varepsilon _r} = 3,\,\,{E_p} = 360\,\pi$$
C
$${\varepsilon _r} = 9,\,\,{E_p} = 360\,\pi$$
D
$${\varepsilon _r} = 9,\,\,{E_p} = 120\,\pi$$
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