Electromagnetic Wave Propagation MCQ Quiz - Objective Question with Answer for Electromagnetic Wave Propagation - Download Free PDF

Calculation:

In a plane electromagnetic wave:

E, B, and the direction of propagation are mutually perpendicular.

The direction of B is determined using the right-hand rule: E × B = direction of wave.

Direction of wave propagation: +x

Electric field E direction: +z

To satisfy +z × ? = +x, the correct direction for B is -y.

Correct Option: (4)

Calculation:

As, Poynting vector

S = E × H

Given: energy transport = negative z-direction

Electric field = positive y-direction

(−k̂) = (+ĵ) × [î]

Hence, according to the vector cross product, the magnetic field should be in the positive x-direction.

Correct option is : (1) Reflected light is completely polarized and the angle of reflection is close to 60°

Using Brewster's law

μ = tan θ_p

⇒ 1.73 = tan θ_p

⇒ √3 = tan θ_p

⇒ θ_p = 60°

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 correct answer is option 4) i.e. Polarization.

CONCEPT:

• A wave is an oscillation that carries energy from one place to another without transporting matter.
• Light is a combination of electric fields and magnetic fields that are perpendicular to each other.
  • So, the two perpendicular planes are occupied by these fields. The electric and magnetic vibrations can simultaneously occur in a number of perpendicular planes. 
  • Therefore, light is an electromagnetic wave. 
  • A wave that oscillates in many numbers of planes is called an unpolarized wave.
  • Using devices called polarizers, light can be made to vibrate along a single plane. Such light waves are called polarized light.
  • Polarization of light occurs when light is reflected, refracted, and scattered. 

​EXPLANATION:

• The direction of vibration of particles is a property associated with waves. Since light shows the vibration of the electric and magnetic field through polarization, the wave nature of light is concluded from polarization.

Additional Information

Concept:

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves.

mathematically, the Poynting vector states that:

\(P = \vec E \times \vec H\;Watt/{m^2}\)

Calculation:

Given,

\({\rm{\vec E}} = \left| {{\rm{\vec E}}} \right|{a_x}\)

 \(\vec H = \left| {\vec H} \right|{a_y}\)

∴  \(P = \left| {{\rm{\vec E}}} \right|\left| {\vec H} \right|.{\vec a_x} \times {\vec a_y}\)

\(= \left| {{\rm{\vec E}}} \right|\left| {\vec H} \right|{\vec a_z}\)

Concept:

1) The direction of the electric field is considered as the polarization of the electromagnetic wave.

2) The direction of propagation of EM wave is given by the cross-product of the vector product of direction of electric field and magnetic field, i.e.

\({\hat a_p} = {\hat a_E} \times {\hat a_H}\)

This is an application of the Poynting theorem.

Analysis:

Given:

\(\vec H = 20\;{e^{ - \alpha x}}\cos \left( {\omega t - 0.25x} \right){\hat a_y}\)

\({\hat a_H} = {\hat a_y}\)

\({\hat a_p} = {\hat a_x}\)

\({\hat a_p} = {\hat a_e} \times {\hat a_H} \Rightarrow {\hat a_x} = {\hat a_E} \times {\hat a_y}\)

\({\hat a_E} = - {\hat a_z}\)

Now, the polarization of the wave = direction of the electric field, i.e.

Wavelength (λ) is equal to the distance traveled by the wave during the time in which any one particle of the medium completes one vibration about its mean position. It is the length of one wave.

Frequency (f) of vibration of a particle is defined as the number of vibrations completed by the particle in one second. It is the number of complete wavelengths traversed by the wave in one second.

The relation between velocity, frequency, and wavelength:

c = f × λ

Where,

c is the speed of light in vacuum = 3 x 10⁸ m/s

\(\lambda = \frac{3\times10^8}{f}\)

But since it is given that the frequency is in MHz, we can write:

\(\lambda = \frac{300\times10^6}{f(MHz)}\)

Since 1 MHz = 10⁶ Hz, the above can be written as:

\(\lambda = \frac{300~MHz}{f(MHz)}​​\)

\(\lambda = \frac{300}{f}\)

Concept:

A uniform plane wave is propagating in the material which is lossless i.e. there is no loss.

Hence R = G = 0, σ = 0 only L and C of the material is considered.

L is represented by μ₀μ_r

And C represented by ε₀ ε_r in the wave.

Propagation velocity, \({V_p} = \frac{\omega }{\beta }\)

And \({V_p} = \frac{C}{{\sqrt {{\mu _r}{\varepsilon _r}} }}\;\;\;\;\;\;\;\left( {C = 3 \times {{10}^8}\;m/s} \right)\)

Calculation:

Now, given that: freq. f = 50 × 10⁶ Hz

Relative permeability μ_r = 2.25

Relative permittivity ε_r = 1

Since the material is lossless, σ = 0

To find β (propagation constant):

\({V_p} = \frac{{3\; \times \;{{10}^8}}}{{\sqrt {2.25\; \times \;1} }} = \frac{{2\pi \left( {50\; \times \;{{10}^6}} \right)}}{\beta }\)

\(\Rightarrow \beta = \frac{\pi }{2}rad/s\)

The Brewster law states the relationship between light waves and polarized light. The polarized light vanishes at this maximum angle.

Brewster angle is the angle when a wave is an incident on the surface of a perfect dielectric at which there is no reflected wave and the incident wave is partially polarized.

The refracted ray is oriented at a 90-degree angle from the reflected ray and is only partially polarized

Brewster angle is given by:

\(\theta = {\tan ^{ - 1}}\sqrt {\frac{{{\epsilon_2}}}{{{\epsilon_1}}}}\)

The British Physicist David Brewster found the relationship between Brewster angle (ip) and refractive index (μ).

μ = tan ip
 

Brewster's Angle-an angle of incidence at which there is no reflection of p-polarized light at an uncoated optical surface.

Brewster Angle:

E_P = parallel polarized component (reflection = 0)

E_S = perpendicular polarized compenent

r = reflection coefficient

τ = transmission coefficient

θ_B = Brewster Angle \(= {\tan ^{ - 1}}\left( {\frac{{{n_2}}}{{{N_1}}}} \right)\)

where n₂ = Refractive index of medium 2)

n₁ = Refractive index of medium (1)

Important Point-

When referring to polarization states, the p-polarization refers to the polarization plane parallel to the polarization axis of the polarizer being used ("p" is for "parallel"). The s-polarization refers to the polarization plane perpendicular to the polarization axis of the polarizer.

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

Poynting vector (S):

It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec S = \vec E \times \vec H\)

S = Poynting vector

E = Electric field and

H = Magnetic field

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves per unit volume.

Application:

Given,

length = l

radious = r

current = i

Potential = V

Since, electric field (E) is the potential per unit length,

Hence, \(E=\frac{V}{l}\)

For a long straight wire conductor, the magnetic field intensity (H) is given by,

\(H=\frac{i}{2\pi r}\)

Hence, the magnitude of pointing vector (S) will be,

S = EH = \(\frac{V}{l}\times \frac{i}{2\pi r}=\frac{Vi}{2\pi rl}\)

CONCEPT:

• When light rays travel from one medium to another then there is bending in the light ray. This phenomenon is called as refraction of light.
• The refraction occurs due to different refractive indexes of the medium for different colours of light.
• When the light ray strikes on a small molecule of any medium particles then it gets spread in different directions then this phenomenon is called as a scattering of light.
• White light coming from the sun consists of seven constituent colours.
• These are – Violet, Indigo, blue, green, yellow, orange and red
• The red colour light has the longest wavelength among them and Violet has the shortest wavelength.

EXPLANATION:

• During the sunrise and sunset time, the light ray coming from the sun travel a longer distance in the atmosphere because the sun is on the horizon.
• The light having the least wavelength will be scattered most and light having the longest wavelength will be scattered least.
• Since the red and orange colour light has maximum wavelength among all other colours so it will be least scattered in the atmosphere and will come directly to our eye.
• As the combination of red and orange colours will come directly to our eyes so the sun appears reddish-orange. This is due to the least scattering by the atmosphere. Hence option 3 is correct.

The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies.

The electromagnetic spectrum can be shown in the figure below:

The wavelength and frequency of different colors are shown in the following table: