Uniform Plane Waves Questions and Answers

Q: The ratio of skin depth in copper at 1 kHz to that at 100 MHz is approximately :

The ratio is about 300.The skin depth \( \delta = \sqrt{\frac{2}{\omega\mu\sigma}} \) varies as \( 1/\sqrt{f} \), so\[ \begin{aligned} \frac{\delta_{1\,\text{kHz}}}{\delta_{100\,\text{MHz}}} &= \sqrt{\frac{10^{8}}{10^{3}}} \\ &= \sqrt{10^{5}} \approx 316 \end{aligned} \]In copper that is about 2 mm at 1 kHz against 6.6 μm at 100 MHz. 30 and 3000 would follow from frequency ratios of \( 10^{3} \) or \( 10^{7} \), not \( 10^{5} \), and 3 is far too small.

Q: Assertion-Reason type questions :Assertion (A) : Electromagnetic waves are transverse in nature.Reason (R) : The electric and magnetic fields are perpendicular to the direction of wave propagation.

The answer is both (A) and (R) are true and (R) is the correct explanation of (A).A wave is transverse when its oscillations are at right angles to the direction in which it travels. In an electromagnetic wave in free space, the electric field E and the magnetic field B are perpendicular to each other and both perpendicular to the direction of propagation, given by \(E \times B\).That perpendicularity of the fields to the direction of propagation is exactly what makes the wave transverse, so the Reason explains the Assertion rather than merely being true alongside it.

Q: If an electromagnetic wave is propagating in a medium with permittivity \(\varepsilon\) and permeability \(\mu\), then \(\sqrt{\frac{\mu}{\varepsilon}}\) is :

\(\sqrt{\mu/\varepsilon}\) is the intrinsic impedance of the medium.For a plane wave, the ratio of the electric to the magnetic field amplitude is \(E/H = \sqrt{\mu/\varepsilon} = \eta\), measured in ohms. In free space it is about 377 Ω.The refractive index is a different combination, \(n = \sqrt{\mu\varepsilon/\mu_{0}\varepsilon_{0}}\), which involves the product, not the ratio. The square root of n has no special role, and energy density depends on the field strengths, not on the material constants alone.

Q: A plane electromagnetic wave is propagating through a perfect dielectric medium with dielectric constant equal to \(\pi\). The phase difference between \(\overline{E}\) and \(\overline{B}\) associated with the wave is :

The phase difference is Zero.In a perfect (lossless) dielectric, Maxwell's equations give \(\overline{B} = \frac{1}{v}\hat{k} \times \overline{E}\) with a real wave speed \(v = 1/\sqrt{\mu\varepsilon}\). The ratio of the fields is real, so E and B oscillate in phase.The value of the dielectric constant (here \(\pi\)) only changes the speed and the ratio of amplitudes, not the phase. A phase lag appears only in a conducting medium, where it approaches \(\pi/4\) for a good conductor.

Q: Velocity of light in air is \(3 \times 10^{8}\) m/s. Velocity of light in a medium having relative permittivity 3 and relative permeability 2 is ........ .

The velocity is \(\sqrt{\frac{3}{2}} \times 10^{8}\) m/s.The speed of light in a medium is \(v = \frac{c}{\sqrt{\epsilon_{r}\mu_{r}}}\):\[ \begin{aligned} v &= \frac{3 \times 10^{8}}{\sqrt{3 \times 2}} \\ &= \frac{3}{\sqrt{6}} \times 10^{8} = \sqrt{\frac{9}{6}} \times 10^{8} \\ &= \sqrt{\frac{3}{2}} \times 10^{8}\ \text{m/s} \end{aligned} \]That is about \(1.22 \times 10^{8}\) m/s, a refractive index of \(\sqrt{6} \approx 2.45\). The option \(\sqrt{2} \times 10^{8}\) uses only one of the two factors, and the forms with \(\sqrt{3}\) in the numerator invert part of the expression.

Q: A square wave guide with perfect conductor boundaries is of cross-sectional area \(l^{2}\). The wave guide is placed in vacuum. The minimum frequency of the electromagnetic waves that can propagate through this wave guide is ........ .

The key's answer is \(\frac{\pi c}{l}\), the cut-off expressed as an angular frequency.In a rectangular guide of sides a and b, the cut-off angular frequency of the \(TE_{mn}\) mode is \(\omega_{c} = c\pi\sqrt{(m/a)^{2} + (n/b)^{2}}\). The lowest mode in a square guide of side l is \(TE_{10}\) (or \(TE_{01}\)):\[ \begin{aligned} \omega_{c} &= \frac{\pi c}{l} \\ f_{c} &= \frac{\omega_{c}}{2\pi} = \frac{c}{2l} \end{aligned} \]Waves below this cut-off cannot propagate. The cycle frequency c/2l is not among the options, so the paper's 'frequency' is the angular frequency. \(\sqrt{2}\pi c/l\) is the cut-off of the higher \(TE_{11}\) and \(TM_{11}\) modes, and \(c/l\) and \(2c/l\) correspond to no lowest mode.

Q: Circular polarization of EM wave means electric and magnetic field must have :

Circular polarization needs two orthogonal linear components with the same magnitude and a phase difference in odd multiples of 90°.If \(E_{x} = E_{0}\cos\omega t\) and \(E_{y} = E_{0}\sin\omega t\), equal in amplitude and 90° apart, the tip of the field vector traces a circle of constant radius E0 as it rotates.With a phase difference of 0 or 180° (even multiples of 90°) the components stay in step and the result is linear polarization. Unequal magnitudes with a 90° difference give elliptical polarization, and merely having non-zero transverse components says nothing about the shape traced.

Q: The ratio of electric field intensity E and magnetic field intensity H has numerical value in free space :

The ratio E/H in free space is about 377 Ω.This ratio is the intrinsic impedance of free space:\[ \begin{aligned} \eta_0 &= \sqrt{\frac{\mu_0}{\varepsilon_0}} \\ &= 120\pi \approx 377\ \Omega \end{aligned} \]75 Ω is the characteristic impedance of common coaxial cable used for television, and 35 Ω and 225 Ω have no special meaning for plane waves in free space.

Q: Skin depth or depth of penetration in a conductor :(1) is inversely proportional to square root of µ and σ(2) is inversely proportional to square root of f(3) is increased as frequency increases(4) is directly proportional to square root of µ and σWhich statements is/are correct ?

Statements 1 and 2 are correct, so the answer is (1) and (2).The skin depth of a good conductor is\[ \delta = \frac{1}{\sqrt{\pi f \mu \sigma}} \]so it is inversely proportional to the square root of μ and σ (1) and to the square root of the frequency f (2).It follows that the skin depth decreases as frequency increases, so statement 3 is false, and it is not directly proportional to the square root of μ and σ, so statement 4 is false. Every option containing 3 or 4 is wrong.

Q: The phase velocity of an electro-magnetic wave depends on frequency in aqueous medium, the said phenomenon is called :

The phenomenon is dispersion.A medium is dispersive when its refractive index, and so the phase velocity of waves in it, depends on frequency. Water is strongly dispersive for electromagnetic waves, so different frequency components travel at different speeds and a pulse spreads out.Polarization is the orientation of the electric field vector, absorption is the loss of wave energy in the medium, and scattering is the redirection of the wave by particles or irregularities. None of these is the frequency dependence of phase velocity.

91.

The ratio of skin depth in copper at 1 kHz to that at 100 MHz is approximately :

A.3
B.30
C.300
D.3000

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: C

The ratio is about 300.

The skin depth \( \delta = \sqrt{\frac{2}{\omega\mu\sigma}} \) varies as \( 1/\sqrt{f} \), so

\[ \begin{aligned} \frac{\delta_{1\,\text{kHz}}}{\delta_{100\,\text{MHz}}} &= \sqrt{\frac{10^{8}}{10^{3}}} \\ &= \sqrt{10^{5}} \approx 316 \end{aligned} \]

In copper that is about 2 mm at 1 kHz against 6.6 μm at 100 MHz. 30 and 3000 would follow from frequency ratios of \( 10^{3} \) or \( 10^{7} \), not \( 10^{5} \), and 3 is far too small.

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92.

Assertion-Reason type questions :

Assertion (A) : Electromagnetic waves are transverse in nature.
Reason (R) : The electric and magnetic fields are perpendicular to the direction of wave propagation.

A.Both (A) and (R) are true and (R) is the correct explanation of (A)
B.Both (A) and (R) are true but (R) is not correct explanation of (A)
C.(A) is true but (R) is false
D.(A) is false and (R) is true

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: A

The answer is both (A) and (R) are true and (R) is the correct explanation of (A).

A wave is transverse when its oscillations are at right angles to the direction in which it travels. In an electromagnetic wave in free space, the electric field E and the magnetic field B are perpendicular to each other and both perpendicular to the direction of propagation, given by \(E \times B\).

That perpendicularity of the fields to the direction of propagation is exactly what makes the wave transverse, so the Reason explains the Assertion rather than merely being true alongside it.

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93.

If an electromagnetic wave is propagating in a medium with permittivity \(\varepsilon\) and permeability \(\mu\), then \(\sqrt{\frac{\mu}{\varepsilon}}\) is :

A.The refractive index of the medium.
B.The square root of the refractive index of the medium.
C.The intrinsic impedance of the medium.
D.The energy density of the wave in the medium.

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: C

\(\sqrt{\mu/\varepsilon}\) is the intrinsic impedance of the medium.

For a plane wave, the ratio of the electric to the magnetic field amplitude is \(E/H = \sqrt{\mu/\varepsilon} = \eta\), measured in ohms. In free space it is about 377 Ω.

The refractive index is a different combination, \(n = \sqrt{\mu\varepsilon/\mu_{0}\varepsilon_{0}}\), which involves the product, not the ratio. The square root of n has no special role, and energy density depends on the field strengths, not on the material constants alone.

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94.

A plane electromagnetic wave is propagating through a perfect dielectric medium with dielectric constant equal to \(\pi\). The phase difference between \(\overline{E}\) and \(\overline{B}\) associated with the wave is :

A.Zero
B.\(\frac{\pi}{4}\)
C.\(\frac{\pi}{2}\)
D.\(\pi\)

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: A

The phase difference is Zero.

In a perfect (lossless) dielectric, Maxwell's equations give \(\overline{B} = \frac{1}{v}\hat{k} \times \overline{E}\) with a real wave speed \(v = 1/\sqrt{\mu\varepsilon}\). The ratio of the fields is real, so E and B oscillate in phase.

The value of the dielectric constant (here \(\pi\)) only changes the speed and the ratio of amplitudes, not the phase. A phase lag appears only in a conducting medium, where it approaches \(\pi/4\) for a good conductor.

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95.

Velocity of light in air is \(3 \times 10^{8}\) m/s. Velocity of light in a medium having relative permittivity 3 and relative permeability 2 is ........ .

A.\(\sqrt{\frac{3}{2}} \times 10^{8}\) m/s
B.\(\frac{\sqrt{3}}{4} \times 10^{8}\) m/s
C.\(\sqrt{2} \times 10^{8}\) m/s
D.\(\frac{\sqrt{3}}{2} \times 10^{8}\) m/s

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: A

The velocity is \(\sqrt{\frac{3}{2}} \times 10^{8}\) m/s.

The speed of light in a medium is \(v = \frac{c}{\sqrt{\epsilon_{r}\mu_{r}}}\):

\[ \begin{aligned} v &= \frac{3 \times 10^{8}}{\sqrt{3 \times 2}} \\ &= \frac{3}{\sqrt{6}} \times 10^{8} = \sqrt{\frac{9}{6}} \times 10^{8} \\ &= \sqrt{\frac{3}{2}} \times 10^{8}\ \text{m/s} \end{aligned} \]

That is about \(1.22 \times 10^{8}\) m/s, a refractive index of \(\sqrt{6} \approx 2.45\). The option \(\sqrt{2} \times 10^{8}\) uses only one of the two factors, and the forms with \(\sqrt{3}\) in the numerator invert part of the expression.

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96.

A square wave guide with perfect conductor boundaries is of cross-sectional area \(l^{2}\). The wave guide is placed in vacuum. The minimum frequency of the electromagnetic waves that can propagate through this wave guide is ........ .

A.\(\frac{c}{l}\)
B.\(\frac{2c}{l}\)
C.\(\frac{\pi c}{l}\)
D.\(\sqrt{2}\frac{\pi c}{l}\)

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: C

The key's answer is \(\frac{\pi c}{l}\), the cut-off expressed as an angular frequency.

In a rectangular guide of sides a and b, the cut-off angular frequency of the \(TE_{mn}\) mode is \(\omega_{c} = c\pi\sqrt{(m/a)^{2} + (n/b)^{2}}\). The lowest mode in a square guide of side l is \(TE_{10}\) (or \(TE_{01}\)):

\[ \begin{aligned} \omega_{c} &= \frac{\pi c}{l} \\ f_{c} &= \frac{\omega_{c}}{2\pi} = \frac{c}{2l} \end{aligned} \]

Waves below this cut-off cannot propagate. The cycle frequency c/2l is not among the options, so the paper's 'frequency' is the angular frequency. \(\sqrt{2}\pi c/l\) is the cut-off of the higher \(TE_{11}\) and \(TM_{11}\) modes, and \(c/l\) and \(2c/l\) correspond to no lowest mode.

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97.

Circular polarization of EM wave means electric and magnetic field must have :

A.Either electric or magnetic non-zero and positive propagation vector
B.Two orthogonal linear components with real magnitude and phase difference in even multiples of 90°
C.Two orthogonal linear components with same magnitude and phase difference in odd multiples of 90°
D.Both components non-zero and perpendicular to the direction of propagation

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

Circular polarization needs two orthogonal linear components with the same magnitude and a phase difference in odd multiples of 90°.

If \(E_{x} = E_{0}\cos\omega t\) and \(E_{y} = E_{0}\sin\omega t\), equal in amplitude and 90° apart, the tip of the field vector traces a circle of constant radius E₀ as it rotates.

With a phase difference of 0 or 180° (even multiples of 90°) the components stay in step and the result is linear polarization. Unequal magnitudes with a 90° difference give elliptical polarization, and merely having non-zero transverse components says nothing about the shape traced.

Discuss in Forum

98.

The ratio of electric field intensity E and magnetic field intensity H has numerical value in free space :

A.35 Ω
B.377 Ω
C.75 Ω
D.225 Ω

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: B

The ratio E/H in free space is about 377 Ω.

This ratio is the intrinsic impedance of free space:

\[ \begin{aligned} \eta_0 &= \sqrt{\frac{\mu_0}{\varepsilon_0}} \\ &= 120\pi \approx 377\ \Omega \end{aligned} \]

75 Ω is the characteristic impedance of common coaxial cable used for television, and 35 Ω and 225 Ω have no special meaning for plane waves in free space.

Discuss in Forum

99.

Skin depth or depth of penetration in a conductor :

(1) is inversely proportional to square root of µ and σ
(2) is inversely proportional to square root of f
(3) is increased as frequency increases
(4) is directly proportional to square root of µ and σ

Which statements is/are correct ?

A.(1) and (3)
B.(1) and (2)
C.(2) and (4)
D.(4) only

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: B

Statements 1 and 2 are correct, so the answer is (1) and (2).

The skin depth of a good conductor is

\[ \delta = \frac{1}{\sqrt{\pi f \mu \sigma}} \]

so it is inversely proportional to the square root of μ and σ (1) and to the square root of the frequency f (2).

It follows that the skin depth decreases as frequency increases, so statement 3 is false, and it is not directly proportional to the square root of μ and σ, so statement 4 is false. Every option containing 3 or 4 is wrong.

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100.

The phase velocity of an electro-magnetic wave depends on frequency in aqueous medium, the said phenomenon is called :

A.Dispersion
B.Polarization
C.Absorption
D.Scattering

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

The phenomenon is dispersion.

A medium is dispersive when its refractive index, and so the phase velocity of waves in it, depends on frequency. Water is strongly dispersive for electromagnetic waves, so different frequency components travel at different speeds and a pulse spreads out.

Polarization is the orientation of the electric field vector, absorption is the loss of wave energy in the medium, and scattering is the redirection of the wave by particles or irregularities. None of these is the frequency dependence of phase velocity.

Discuss in Forum

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