Phase Velocity MCQ Quiz - Objective Question with Answer for Phase Velocity - Download Free PDF

Concept:

Wave Equation:

The general form of a wave equation is y = A sin(kx - ωt), where:

A = Amplitude of the wave

k = Wave number (k = 2π / λ)

ω = Angular frequency (ω = 2πf)

t = Time

x = Position

The wave velocity v can be calculated using the relation:

v = ω / k

Calculation:

Given the wave equation: y = 0.5 sin(2π / λ (400t - x)) m, we can identify the following:

ω = 2π × 400 = 800π rad/s

k = 2π / λ

The velocity of the wave is given by:

v = ω / k = (800π) / (2π / λ) = 400λ

Since the equation is in the standard wave form, we can conclude that the velocity of the wave is 400 m/s.

∴ The velocity of the wave is 400 m/s, which corresponds to Option 3.

Calculation:

We start with the given dispersion relation:

\(\frac{\omega^{2}}{k^{2}}=\frac{T}{\mu}+\alpha k^{2}\)

Step 1: Expression for Phase Velocity

The phase velocity $V_P$ is defined as:

\(V_p=\frac{\omega}{k}\)

Rearranging the given dispersion relation,

\(\omega^2=k^2(\frac{T}{\mu}+\alpha k^2)\)

Taking the square root,

\(\omega=k\sqrt{\frac{T}{\mu}+\alpha k^2}\)

For small \(\alpha k^2\), we can use the binomial expansion:

\(V_p=\sqrt{\frac{T}{\mu}}\sqrt{1+\frac{\alpha k^2\mu}{T}}\)

Using \(\sqrt{1+x}\approx 1+\frac{x}{2}\) for small x,

\(V_p\approx\sqrt{\frac{T}{\mu}}({1+\frac{\alpha k^2\mu}{T}})\)

Thus, option '2' is correct.

FORMULA:

\(\upsilon p = \frac{\omega }{\beta }\)

Where,

υp = phase velocity

ω = frequ

ency

β = Phase constant

Solution:

Given:

\(\vec E = {E_0}\;{e^{J\left( {\omega t + 3x - 4y} \right)}} \cdot \;\frac{{8\;\widehat {{a_x}} + 6\;\widehat {{a_y}} + 5\;\widehat {{a_2}}}}{{\sqrt {125} }}\;V/m\)

Comparing it with standard plane wave equation

\({E_0} = {e^{J\left( {\omega t - \vec k \cdot \vec r} \right)}}\)

K = -3x + 4y = +βx αx + βy αy

Βy = 4

\(\upsilon p = \frac{\omega }{\beta }\)

∵ f = 10 GHg = 10¹⁰ Hg

\(\upsilon g = \frac{\omega }{{By}} = \frac{{2\pi f}}{4} = \frac{{2\pi \times {{10}^{10}}}}{4}\)

\( = 1056 \times {10^{10}}\;m/sec\)

The phase velocity of any mode is greater than the speed of light in free space in a rectangular waveguide.

Explanation:

For a wave propagating in an air-filled rectangular waveguide, the guided wavelength is given by:

\({\lambda _g} = \frac{{2\pi }}{{\sqrt {{k^2} - k_c^2} }}\)                                    

\({\lambda _o} = \frac{{2\pi }}{k}\) and \({\lambda _c} = \frac{{2\pi }}{{{k_c}}}\)                                       

\(= \frac{{2\pi }}{{k\sqrt {1 - {{\left( {\frac{{{k_c}}}{k}} \right)}^2}} \;}}\)

\(\beta = \sqrt {k - k_c^2} \)   

\(= \frac{{{\lambda _o}}}{{\sqrt {1 - {{\left( {\frac{{{\lambda _o}}}{{{\lambda _c}}}} \right)}^2}} }}\)

Hence λg is always greater than λo

wave impedance for TE mode is kη/β and for TM and mode \(\frac{\beta }{k}\eta \)

Since β ≤ k wave impedance can be less for TM modes.

Phase velocity \({v_p} = \frac{\omega }{\beta }\)

\(= \frac{\omega }{{\sqrt {{k^2} - k_c^2} }}\)

Which is always greater than the free space light speed ω/k

Concept:

The phase velocity (Vp) of a plane electromagnetic wave is given as-

\({V_p} = \frac{1}{{\sqrt {\mu \varepsilon } }}\)

Here, μ = μoμr

ε = εo εr

\({v_p} = \frac{1}{{\sqrt {{\mu _o}{\mu _r}{\varepsilon _o}{\varepsilon _r}} }}\)

\({v_p} = \frac{c}{{\sqrt {{\mu _r}{\varepsilon _r}} }}\)

\(c = \frac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }} = 3 \times {10^8}\;m/sec\)

μ0 = permeability in free space 4π × 10-7 H/m

εo = permittivity in free space 8.854 × 10-12 C2/Nm2

Calculation:

For the medium in question,

µr = 1.5, εr = 6

\(v_p = \frac{{3 \times {{10}^8}}}{{\sqrt {1.5 \times 6} }}\)

Concept:

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes known as the modulation or envelope of the wave propagates through space.

The group velocity is defined by the equation:

\({v_g} = \frac{{d\omega }}{{dk}}\)

Where ω = wave’s angular frequency

k = angular wave number = 2π/λ 

Wave theory tells us that a wave carries its energy with the group velocity. For matter waves, this group velocity is the velocity u of the particle.

Energy of a photon is given by the planck as:

E = hν

With ω = 2πν

ω = 2πE/h      ----- (1)

Wave number is given by:

k = 2π/λ = 2πp/h    ----(2)

where λ = h/p (de broglie)

Now from equations 1 and 2, we get:

\(d\omega = \frac{{2\pi }}{h}dE;\)

\(dk = \frac{{2\pi }}{h}dp;\)

\(\frac{{d\omega }}{{dk}} = \frac{{dE}}{{dp}}\)

By definition: \({v_g} = \frac{{d\omega }}{{dk}}\)

v_g = dE/dp   ---- (3)

If a particle of mass m is moving with a velocity v, then

\(E = \frac{1}{2}m{v^2} = \frac{{{p^2}}}{{2m}}\)

\(\frac{{dE}}{{dp}} = \frac{p}{m} = {v_p}\)    ---(4)

Now from equations 3 and 4:

v_g = v_p

Phase velocity:

It is the rate at which the phase of the wave propagates in space. Mathematically:

\({V_p} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }}\)

ω_c = Cut off frequency

ω = Frequency of operation

Group velocity:

It is the velocity at which the overall envelope of the wave travels.

\({V_g} = c\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} \)

Multiplying both velocities, we get:

\({V_p}{V_g} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }} \times c\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} \)

\({V_p}{V_g} = {c^2}\)

Concept:

The general equation of magnetic field intensity is given by:

E = E0 cos (ωt - βz)ax A / m

The phase velocity is defined as the rate at which the phase of the wave propagates in space.

Mathematically this is calculated as:

\({v_p} = \frac{ω }{β }=\frac{2\pi f}{\frac{2\pi}{λ}}=fλ\)

Calculation:

Given:

f = 5.0 GHz, λ = 3.0 cm

So, the phase velocity of the wave in the medium will be:

v_p = fλ 

v_p = 5 x 10⁹ x 3 x 10^-2

v_p = 1.5 x 10⁸ m/sec

FORMULA:

\(\upsilon p = \frac{\omega }{\beta }\)

Where,

υp = phase velocity

ω = frequ

ency

β = Phase constant

Solution:

Given:

\(\vec E = {E_0}\;{e^{J\left( {\omega t + 3x - 4y} \right)}} \cdot \;\frac{{8\;\widehat {{a_x}} + 6\;\widehat {{a_y}} + 5\;\widehat {{a_2}}}}{{\sqrt {125} }}\;V/m\)

Comparing it with standard plane wave equation

\({E_0} = {e^{J\left( {\omega t - \vec k \cdot \vec r} \right)}}\)

K = -3x + 4y = +βx αx + βy αy

Βy = 4

\(\upsilon p = \frac{\omega }{\beta }\)

∵ f = 10 GHg = 10¹⁰ Hg

\(\upsilon g = \frac{\omega }{{By}} = \frac{{2\pi f}}{4} = \frac{{2\pi \times {{10}^{10}}}}{4}\)

\( = 1056 \times {10^{10}}\;m/sec\)

Concept:

The phase velocity (Vp) of a plane electromagnetic wave is given as-

\({V_p} = \frac{1}{{\sqrt {\mu \varepsilon } }}\)

Here, μ = μoμr

ε = εo εr

\({v_p} = \frac{1}{{\sqrt {{\mu _o}{\mu _r}{\varepsilon _o}{\varepsilon _r}} }}\)

\({v_p} = \frac{c}{{\sqrt {{\mu _r}{\varepsilon _r}} }}\)

\(c = \frac{1}{{\sqrt {{\mu _o}{\varepsilon _o}} }} = 3 \times {10^8}\;m/sec\)

μ0 = permeability in free space 4π × 10-7 H/m

εo = permittivity in free space 8.854 × 10-12 C2/Nm2

Calculation:

For the medium in question,

µr = 1.5, εr = 6

\(v_p = \frac{{3 \times {{10}^8}}}{{\sqrt {1.5 \times 6} }}\)

Concept:

Wave Equation:

The general form of a wave equation is y = A sin(kx - ωt), where:

A = Amplitude of the wave

k = Wave number (k = 2π / λ)

ω = Angular frequency (ω = 2πf)

t = Time

x = Position

The wave velocity v can be calculated using the relation:

v = ω / k

Calculation:

Given the wave equation: y = 0.5 sin(2π / λ (400t - x)) m, we can identify the following:

ω = 2π × 400 = 800π rad/s

k = 2π / λ

The velocity of the wave is given by:

v = ω / k = (800π) / (2π / λ) = 400λ

Since the equation is in the standard wave form, we can conclude that the velocity of the wave is 400 m/s.

∴ The velocity of the wave is 400 m/s, which corresponds to Option 3.

The phase velocity of any mode is greater than the speed of light in free space in a rectangular waveguide.

Explanation:

For a wave propagating in an air-filled rectangular waveguide, the guided wavelength is given by:

\({\lambda _g} = \frac{{2\pi }}{{\sqrt {{k^2} - k_c^2} }}\)                                    

\({\lambda _o} = \frac{{2\pi }}{k}\) and \({\lambda _c} = \frac{{2\pi }}{{{k_c}}}\)                                       

\(= \frac{{2\pi }}{{k\sqrt {1 - {{\left( {\frac{{{k_c}}}{k}} \right)}^2}} \;}}\)

\(\beta = \sqrt {k - k_c^2} \)   

\(= \frac{{{\lambda _o}}}{{\sqrt {1 - {{\left( {\frac{{{\lambda _o}}}{{{\lambda _c}}}} \right)}^2}} }}\)

Hence λg is always greater than λo

wave impedance for TE mode is kη/β and for TM and mode \(\frac{\beta }{k}\eta \)

Since β ≤ k wave impedance can be less for TM modes.

Phase velocity \({v_p} = \frac{\omega }{\beta }\)

\(= \frac{\omega }{{\sqrt {{k^2} - k_c^2} }}\)

Which is always greater than the free space light speed ω/k

Concept:

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes known as the modulation or envelope of the wave propagates through space.

The group velocity is defined by the equation:

\({v_g} = \frac{{d\omega }}{{dk}}\)

Where ω = wave’s angular frequency

k = angular wave number = 2π/λ 

Wave theory tells us that a wave carries its energy with the group velocity. For matter waves, this group velocity is the velocity u of the particle.

Energy of a photon is given by the planck as:

E = hν

With ω = 2πν

ω = 2πE/h      ----- (1)

Wave number is given by:

k = 2π/λ = 2πp/h    ----(2)

where λ = h/p (de broglie)

Now from equations 1 and 2, we get:

\(d\omega = \frac{{2\pi }}{h}dE;\)

\(dk = \frac{{2\pi }}{h}dp;\)

\(\frac{{d\omega }}{{dk}} = \frac{{dE}}{{dp}}\)

By definition: \({v_g} = \frac{{d\omega }}{{dk}}\)

v_g = dE/dp   ---- (3)

If a particle of mass m is moving with a velocity v, then

\(E = \frac{1}{2}m{v^2} = \frac{{{p^2}}}{{2m}}\)

\(\frac{{dE}}{{dp}} = \frac{p}{m} = {v_p}\)    ---(4)

Now from equations 3 and 4:

v_g = v_p

Phase velocity:

It is the rate at which the phase of the wave propagates in space. Mathematically:

\({V_p} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }}\)

ω_c = Cut off frequency

ω = Frequency of operation

Group velocity:

It is the velocity at which the overall envelope of the wave travels.

\({V_g} = c\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} \)

Multiplying both velocities, we get:

\({V_p}{V_g} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }} \times c\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} \)

\({V_p}{V_g} = {c^2}\)

Concept:

The general equation of magnetic field intensity is given by:

E = E0 cos (ωt - βz)ax A / m

The phase velocity is defined as the rate at which the phase of the wave propagates in space.

Mathematically this is calculated as:

\({v_p} = \frac{ω }{β }=\frac{2\pi f}{\frac{2\pi}{λ}}=fλ\)

Calculation:

Given:

f = 5.0 GHz, λ = 3.0 cm

So, the phase velocity of the wave in the medium will be:

v_p = fλ 

v_p = 5 x 10⁹ x 3 x 10^-2

v_p = 1.5 x 10⁸ m/sec