EM Waves

A uniform plane wave with electric field $\vec{E}(x)=A_y \hat{a}_y e^{-j \frac{2 \pi x}{3}} \mathrm{~V} / \mathrm{m}$ is travelling in the air (relative permittivity, $\dot{o}_r=1$ and relative permeability, $\mu_r=1$ )...

A transparent dielectric coating is applied to glass ($$\varepsilon _r=4,\mu_r=1$$) to eliminate the reflection of red light ($$\lambda_0=0.75~\mu\mathrm{m}$$). The minimum thickness of the dielectric coating, in $$\mu\m...

The electric field of a plane electromagnetic wave is $$E = {a_x}{C_{1x}}\cos (\omega t - \beta z) + {a_y}{C_{1y}}\cos (\omega t - \beta z + \theta )$$ V/m. Which of the following combination(s) will give rise to a left...

Consider a narrow band signal, propagating in a lossless dielectric medium $$(\varepsilon_r=4,\mu_r=1)$$, with phase velocity $$v_p$$ and group velocity $$v_g$$. Which of the following statement is true? ($$c$$ is the ve...

Consider the following wave equation, $${{{\partial ^2}f(x,t)} \over {\partial {t^2}}} = 10000{{{\partial ^2}f(x,t)} \over {\partial {x^2}}}$$ Which of the given options is/are solution(s) to the given wave equation?

The refractive indices of the core and cladding of an optical fiber are 1.50 and 1.48 respectively. The critical propagation angle. Which is defined as the maximum angle that the light beam makes with the axis of the opt...

Consider a rectangular coordinate system $(x, y, z)$ with unit vectors $a_x, a_y$ and $a_z$. A plane wave traveling in the region $z \geq 0$ with electric field vector $E=10 \cos \left(2 \times 10^8 t+\beta z\right) a_y$...

The magnetic field of a uniform plane wave in vacuum is given by $$ \vec{H}(x, y, z, t)=\left(\hat{a}_x+2 \hat{a}_y+b \hat{a}_z\right) \cos (\omega t+3 x-y-z) . $$ The value of $b$ is $\_\_\_\_$ .

The expression for an electric field in free space is $$E = {E_0}\left( {\widehat x + \widehat y + j2\widehat z} \right){e^{ - j\left( {\omega t - kx + ky} \right)}},$$ where $$x,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\...

An optical fiber is kept along the$$\widehat Z$$ direction. The refractive indices for the electric fields along $$\widehat X$$ and $$\widehat Y$$ directions in the fiber are $${n_x} = 1.5000$$ and $${n_y} = 1.5001$$, re...

The electric field of a uniform plane wave travelling along the negative $$z$$ direction is given by the following equation: $$$\overrightarrow E {}_w^i = \left( {{{\widehat a}_{_x}} + j{{\widehat a}_{_y}}} \right){E_0}{...

If a right-handed circularly polarized wave is incident normally on a plane perfect conductor, then the reflected wave will be

Consider the time-varying vector $$I = \,\hat x\,\,15\,\cos \,(\omega \,t) + \,\hat y\,5\,sin(\omega \,t)$$ in Cartesian coordinates, where $$\omega $$ > 0 is a constant. When the vector magnitude $$\left| I \right|$$ is...

Let the electric field vector of a plane electromagnetic wave propagating in a homogenous medium be expressed as $$E = \widehat x{E_x}\,{e^{ - j\left( {wt - \beta z} \right)}},$$ , where the propagation constant $$\beta...

The electric field component of a plane wave traveling in a lossless dielectric medium is given by $$$\overrightarrow E \left( {z,t} \right) = {\widehat a_y}2\cos \left( {{{10}^8}t - {z \over {\sqrt 2 }}} \right)V/m.$$$...

The electric field of a uniform plane electromagnetic wave is $$$\vec E = \left( {{{\overrightarrow a }_x} + j4{{\overrightarrow a }_y}} \right)\exp \left[ {j\left( {2\pi \times {{10}^7}t - 0.2z} \right)} \right]$$$ The...

The electric field intensity of a plane wave traveling in free space is given by the following expression $$E\left( {x,t} \right) = {\widehat a_{_y}}24\pi \,\,\cos \left( {\omega t - {k_0}x} \right)\,\,\,\left( {V/m} \ri...

The electric field intensity of a plane wave propagating in a lossless non-magnetic medium is given by the following expression $$\overrightarrow E \left( {z,t} \right) = {\widehat a_x}5\cos \left( {2\pi \times {{10}^9}t...

Assume that a plane wave in air with an electric field $$\overrightarrow E = 10\cos \left( {\omega t - 3x - \sqrt {3z} } \right){\widehat a_{_y}}\,\,\,V/m$$ is incident on a non-magnetic dielectric slab of relative permi...

For an antenna radiating in free space, the electric field at a distance of 1 km is found to be 12 m V/m. Given that intrinsic impedance of the free space is 120$$\pi \Omega $$, the magnitude of average power density due...

If the electric field of a plane wave is $$$\overrightarrow E \left( {z,t} \right) = \widehat x3\cos \left( {\omega t - kz + {{30}^ \circ }} \right) - \widehat y4\sin \left( {\omega t - kz + {{45}^ \circ }} \right)\left(...

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...

The electric field of a uniform plane electromagnetic wave in free spce, along the positive x direction, is given by $$\vec E = 10\left( {{{\widehat a}_y} + j{{\widehat a}_z}} \right){e^{ - j25x}}.$$ The frequency and po...

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 =...

A plane wave having the electric field component $$${\overrightarrow E _i} = 24\,\,\cos \,\,\left( {3 \times {{10}^8}\,t - \beta \,y} \right){\widehat a_z}\,\,V/m$$$ and traveling in free space is incident normally on a...

A uniform plane wave in the free space is normally incident on an infinitely thick dielectric slab (dielectric constant $${\varepsilon _r} = 9$$ ). The magnitude of the reflection coefficient is

A plane wave of wavelength $$\lambda $$ is traveling in a direction making an angle $${{{30}^ \circ }}$$ with positive $$x$$-axis and $${{{90}^ \circ }}$$ with positiv $$y$$-axis. The $$\overrightarrow E $$ field of the...

The $$\overrightarrow H $$ field (in A/m) of a plane wave propagating in free space is given by $$$\overrightarrow H = \widehat x{{5\sqrt 3 } \over {{\eta _0}}}\cos \left( {\omega \,t - \beta \,z} \right) + \widehat y{5...

When a plane wave traveling in free-space is incident normally on a medium having $${\varepsilon _r} = 4.0,$$ the fraction of power transmitted into the medium is given by

The electric field of an electromagnetic wave propagating in the positive z-direction is given by $$$E = {\widehat a_x}\sin \left( {\omega t - \beta z} \right) + {\widehat a_y}\sin \left( {\omega t - \beta z + \pi /2} \r...

A medium of relative permittivityb $${\varepsilon _r} = 2$$ forms an interface with free-space. A point source of electromagnetic energy is located in the medium at a depth of $$1$$ meter from the interface. Due to the t...

The magnetic field intensity vector of a plane wave is given by $$\overline H \left( {x,y,z,t} \right) = 10\,\sin \left( {50000t + 0.004x + 30} \right){\mathop a\limits^ \cap _y}$$ Where $${\mathop a\limits^ \cap _y}$$ d...

Refractive index of glass is 1.5. Find the wavelength of a beam of light with a frequency of $${10^{14}}$$ Hz in glass. Assume velocity of light is $$3\,\, \times \,{10^8}$$ m/s in vacuum.

A plane electromagnetic wave propagating in free space in incident normally on a large slab of loss-less, non-magnetic, dielectric material with $$\varepsilon > {\varepsilon _0}$$. Maxima and minima are observed when the...

If the electric field intensity associated with a uniform plane electromagnetic wave traveling in a perfect dielectric medium is given by $$E\left( {z,\,t} \right) = \,10\,\cos \left( {2\pi \times {{10}^7}\,\,t - 0.1\,\,...

A uniform plane wave traveling in air is incident on the plane boundary between air and another dielectric medium with $${\varepsilon _r} = 4$$. The reflection coefficient for the normal incidence, is

The depth of penetration of electromagnetic wave in a medium having conductivity $$\sigma $$ at a frequency of 1 KHz is 25 cm. The depth of penetration at a frequency of 4 KHz will be

A plane wave is characterized by $$$\overrightarrow E = \left( {0.5\mathop x\limits^ \cap + \mathop y\limits^ \cap \,{e^{j\pi /2}}} \right){e^{j\omega t - jkz}}.$$$ This wave is

The line-of-sight communication requires the transmit and receive antennas to face each other. If the transmit antenna is vertically polarized for best reception the receiver antenna should be

A person with a receiver is 5 Km away from the transmitter. What is the distance that this person must move further to detect a 3-dB decrease in signal strength?

Distilled water at $${25^ \circ }C$$ is characterized by $$\sigma = 1.7 \times {10^{ - 4}}$$ mho/m and $$ \in = 78{ \in _0}$$ at a frequency of $$3 GHz$$. Its loss tangent $$\tan \delta $$ is

A medium has breakdown strength of $$16$$ KV/m r.m.s. Its relative permeability is $$1.0$$ and relative permittivity is $$4.0$$. A plane electromagnetic wave is transmitted through the medium. Calculate the maximum possi...

A uniform plane electromagnetic wave incident normally on a plane surface of a dielectric material is reflected with a VSWR of 3. What is the percentage of incident power that is reflected?

If a plane electromagnetic wave satisfies the equation $${{{\partial ^2}\,{E_x}} \over {\partial \,{z^2}}} = \,{c^2}{{{\partial ^2}\,{E_x}} \over {\partial \,{t^2}}},$$ the wave propagates in the

A uniform plane wave in air impinges at 45° angle on a lossless dielectric material with dielectric constant $${\varepsilon _r}$$. The transmitted wave propagates in a 30° direction with respect to the normal. The value...

A TEM wave is incident normally upon a perfect conductor. The E and H fields at the boundary will be respectively

Identify which one of the following will NOT satisfy the wave equation.

The depth of penetration of a wave in a lossy dielectric increases with increasing

The electric field vector of a wave is given as $$$\vec E = {E_0}{\mkern 1mu} {e^{j\left( {\omega t + 3x - 4y} \right)}}{\mkern 1mu} {{8{{\vec a}_x} + 6{{\vec a}_y} + 5{{\vec a}_z}} \over {\sqrt {125} }}\,\,V/m$$$ Its fr...

The wavelength of a wave with propagation constant $$\left( {0.1\,\pi + j\,0.2\pi } \right){m^{ - 1}}$$ is

The time averaged Poynting vector, in W/m 2 , for a wave with $$\vec E = 24{e^{j\left( {\omega t + \beta z} \right)}}{\mkern 1mu} {\overrightarrow a _y}$$ V/m in free space is

The polarization of a wave with electric field vector $$\overrightarrow E = {E_0}\,{e^{j\left( {\omega t - \beta z} \right)}}\left( {\overrightarrow {{a_x}} + \overrightarrow {{a_y}} } \right)$$ is

A plane wave with $$\overrightarrow E = 10\,{e^{j\left( {\omega t - \beta z} \right)\,}}\,\,{\overrightarrow a _{_y}}$$ is incident normally on a thick plane conductor lying in the $$X - Y$$ plane. Its conductivity is $$...

The intrinsic impedance of copper at high frequencies is

A uniform plane wave in air is normally incident on an infinitely thick slab. If the reflective index of the glass slab is 1.5, then the percentage of the incident power that is reflected from the air-glass interface is

A transverse electromagnetic wave with circular polarization is received by a dipole antenna. Due to polarization mismatch, the power transfer efficiency from the wave to the antenna is reduced to about

Some unknown material has a conductivity of $${10^{ 6}}$$ $$mho/m$$ and a permeability of $$4\pi \times {10^{ - 7}}\,\,\,\,\,H/m.$$ The skin depth for the material at $$1GHz$$ is

The intrinsic impedance of a lossy dielectric medium is given by

Two dipoles are so fed and oriented in free space that they produce the following electromagnetic waves: $$${E_x} = 10\,{e^{j\left( {\omega t - z\pi /3} \right)}}\,\,volts/metre$$$ $$${E_y} = j\,\,10\,{e^{j\left( {\omega...

A plane electromagnetic wave traveling along the +z-direction, has its electric field given by E x = 2 cos $$\left( {\omega t} \right)$$ and E y = 2 cos $$\left( {\omega t + {{90}^ \circ }} \right)$$ the wave is

A wave traveling in the $$+Z$$-direction, is the resultant of two linearly polarized components, $${E_x}\,\,\, = \,\,\,\,\,\,3$$ $$\,\,\,\,\cos \omega t$$ and $$\,{E_y}\,\,\,\, = \,\,\,2\,\cos \,\left( {\omega t + {{45}^...

A material is described by the following electrical parameters at a frequency of $$10$$ GHz is $$\sigma = {10^6}$$ mho/m, $$\mu = {\mu _0},$$ and $$ \in /{ \in _0} = 10.$$ The material at this frequency is considered to...

A uniform plane wave traveling in free space along the +$$z$$ direction and having its electric field along the$$ x$$-direction, is normally incident on a thick brass sheet, infinite in extent (along the $$x$$-and $$y$$-...

A uniform plane electromagnetic wave traveling in free-space enters into a lossless medium at normal incidence. In the medium its velocity reduces by 50% and in free space sets up a standing wave having a reflection coef...

The electric field component of a uniform plane electromagnetic wave propagating in the $$Y$$-direction in a lossless medium will satisfy the equation

A 300 MHz plane E.M. wave is propagating in free space. The wave is incident normally on an infinite copper slab. For the transmitted wave in the copper slab, calculate the following (a) Attenuation constant (b) Phase co...

The incoming solar radiation at a place on the surface of the earth is 1.2 KW/m 2 .The amplitude of the electric field corresponding to this incident power is nearly equal to

The skin - depth of copper at a frequency of $$3 GHz$$ is $$1$$ micron ($${{{10}^{ - 6}}}$$ metre). At $$12 GHz$$, for a non - magnetic conductor whose conductivity is $$1/9$$ times that of copper, the skin $$-$$ depth w...

In the case of ground wave propagation, at far-off distances from the transmitting tower antenna, the Poynting vector is

In a good conductor the phase relation between the tangential components of electric field E t and the magnetic field H t is as follows

For an electromagnetic wave incident from one medium to a second medium, total reflection takes place when