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

Solution:

We are given the equation for the electric field of an electromagnetic wave in free space:

E = 10 cos(10⁷t + kx) V/m, where t is in seconds and x is in meters.

By comparing the given equation with the standard form of the equation for a wave, we can deduce the following:

The amplitude E₀ = 10 V/m.

The angular frequency ω = 10⁷ rad/s.

Now, the wave velocity (c) is related to the wavelength (λ) by the equation:

c = vλ / 2π

Solving for λ (wavelength):

λ = 2πc / ω = (2π * 3 × 10⁸) / 10⁷ = 188.4 m

Next, the wave number k is related to the wavelength by the formula:

k = ω / c = 10⁷ / (3 × 10⁸) = 0.033 rad/m

Thus, the wavelength λ is 188.4 m and the wave number k is 0.033 rad/m. The amplitude is 10 V/m.

Additionally, the wave propagates along the y-direction (as indicated by the positive sign in the argument of the cosine function), not along the x-direction as stated in the options.

Therefore, the correct pair of statements are: (A) and (C).

Concept:

The speed of an electromagnetic wave in a medium is related to the permittivity (ε) and permeability (μ) of the medium by the equation:

v = 1 / √(ε₀ εr μ₀ μr),

where:

• v is the speed of the wave in the medium,
• ε₀ is the permittivity of free space,
• μ₀ is the permeability of free space,
• εr is the relative permittivity of the medium,
• μr is the relative permeability of the medium.

For free space, ε₀ = 8.854 × 10⁻¹² F/m and μ₀ = 4π × 10⁻⁷ H/m.

Calculation:

Given, speed of the wave, v = 2 × 10⁸ m/s and μr = 1.

Using the relation for wave speed:

v = 1 / √(ε₀ εr μ₀ μr)

Substitute the known values:

2 × 10⁸ = 1 / √(8.854 × 10⁻¹² × εr × 4π × 10⁻⁷ × 1)

2 × 10⁸ = 1 / √(1.112 × 10⁻¹⁸ × εr)

Squaring both sides:

(2 × 10⁸)² = 1 / (1.112 × 10⁻¹⁸ × εr)

4 × 10¹⁶ = 1 / (1.112 × 10⁻¹⁸ × εr)

εr = (1.112 × 10⁻¹⁸) / (4 × 10¹⁶)

εr = 2.25

∴ The relative permittivity εr is 2.25. Option 3) is correct.

Explanation:

Dispersion occurs when the phase velocity of an electromagnetic wave depends on its frequency. This causes different frequency components of a wave to travel at different speeds in the medium, leading to phenomena such as the separation of white light into its constituent colors in a prism.

Thus, option '2' is correct.

Calculation:

We are given that an electromagnetic wave traveling through free space is described by the electric field E and the magnetic field B as follows:

E = E₀ sin(kx - ωt)

B = B₀ sin(kx - ωt)

We need to determine the correct relationship between the amplitudes E₀ and B₀.

For an electromagnetic wave traveling through free space, the magnitudes of the electric field and magnetic field are related by the speed of light c:

c = E₀ / B₀

where c is the speed of light in free space.

We also know that for a wave, the wave number k and the angular frequency ω are related to the speed of light by:

c = ω / k

Combining these two equations, we get:

E₀ / B₀ = ω / k

Rearranging this equation, we find the relationship:

E₀ k = B₀ ω

Therefore, the correct answer is:

Option 1: E₀ k = B₀ ω

Final Answer: The correct relationship is E₀ k = B₀ ω, which corresponds to Option 1.

Concept: 

The intensity I of an electromagnetic wave is related to the peak electric field E_o by the formula:

\(I=\frac{1}{2} \varepsilon_0 E_0^2 c\)

Calculation: 

⇒ I = \(\frac{1}{2} 8.5 \times 10^{-12} \times(56.5)^2 \times 3 \times 10^8\)

⇒ I = 4.24 W/m2

∴ the Correct answer is Option (2): 4.24 Wm–2

The correct answer is Distance.

Concept:

Light year:

• The distance travelled by light in one year is called as light year.
•  It is the unit of astronomical distance.
• 1 light year = 9.461 trillion kilometers which is a very large distance.

Explanation:

• As light year is the distance travelled by light in a year so light year is a unit of distance. Hence option 2 is correct.

Important points:

• We can measure a very large distance with the help of light year. Thus light year is a unit for measurement of very large distance.

CONCEPT:

• The electric field is always perpendicular to the magnetic field, and both fields are directed at right-angles to the direction of propagation of the wave.
• The wave propagates in the direction E × B. Electromagnetic waves are a type of transverse wave.

EXPLANATION:

• In electromagnetic waves the electric field is always perpendicular to the magnetic field and both field are directed at right –angles to the direction of propagation of the wave.

So option 4 is correct.

CONCEPT:

Electromagnetic waves or EM waves:

• The waves that are formed as a result of vibrations between an electric field and a magnetic field and they are perpendicular to each other and to the direction of the wave is called an electromagnetic wave.
• The accelerating charged particle produces an electromagnetic (EM) wave.
• A charged particle oscillating about an equilibrium position is an accelerating charged particle.
• Electromagnetic waves do not require any matter to propagate from one place to another as it consists of photons. They can move in a vacuum.

Properties of electromagnetic waves:

• Not have any charge or we can say that they are neutral.
• Propagate as a transverse wave.
• They move with the velocity the same as that of light.
• The speed of all the electromagnetic waves is 3×108 m/s in a vacuum.
• It contains energy and they also contain momentum.
• They can travel in a vacuum also.

EXPLANATION:

• We know that x-rays, ultraviolet rays, and gamma rays are electromagnetic waves.
• The speed of all the electromagnetic waves is 3 × 108 m/s in a vacuum. Hence, option 4 is correct.

CONCEPT:

• Electromagnetic waves or EM waves: The waves that are formed as a result of vibrations between an electric field and a magnetic field and they are perpendicular to each other and to the direction of the wave is called an electromagnetic waves.
• The accelerating charged particle produces an electromagnetic (EM) wave.
• A charged particle oscillating about an equilibrium position is an accelerating charged particle.
• Electromagnetic waves do not require any matter to propagate from one place to another as it consists of photons. They can move in a vacuum.

Properties of electromagnetic waves:

• The electromagnetic waves are neutral.
• Propagate as a transverse wave.
• They move with a velocity the same as that of light i.e 3 × 108 m/s.
• It contains energy and also contains momentum.
• They can travel in a vacuum also.

Characteristics of Electromagnetic waves:

• The accelerated charge particles are the source of electromagnetic waves.
• Electromagnetic waves are transverse in nature as they propagate by varying the electric and magnetic fields such that the two fields are perpendicular to each other.
• Electromagnetic waves have constant velocity in vacuum 
• Electromagnetic wave propagation does not require any material medium to travel.
• Electromagnetic waves follow the principle of superposition.
• Electromagnetic waves are not reflected by an electric or magnetic field.

Explanation:

• Hence we can see that Moving charges are not responsible to produce electromagnetic waves,
• Instead Accelerated charges are responsible to produce electromagnetic waves.

CONCEPT: 

• Electromagnetic spectrum: It is a collection of a range of different waves in sequential order from radio to gamma electromagnetic waves.

Properties of X-ray:

• X-rays are electromagnetic waves with wavelength range 0. 1Å – 100Å.
• The wavelength of X-rays is small in comparison to the wavelength of the light. Hence they carry more energy.
• They are invisible.
• They travel in a straight line with the speed of light.
• X-rays are measured in Rontgen (a measure of ionization power).
• X-rays carry no charge so they are not reflected in the magnetic field and electric field.
• They ionize gases.
• They produce photoelectric effect and Compton effect
• These cannot be used in Radar because they are not reflected by the target.
• They show all the important properties of light rays like reflection, refraction, interference, diffraction, and polarization, etc.

​EXPLANATION:

• From above it is clear that due to the high penetrating power of X – rays, it shows all the important properties of light rays like reflection, refraction, interference, diffraction, and polarization, etc. Therefore option 4 is correct.

The correct answer is option 3) i.e. Velocity

CONCEPT:

• Electromagnetic spectrum: It is the arrangement of electromagnetic radiations according to their frequency or wavelength.
  • ​The electromagnetic spectrum has a range of frequencies, wavelengths, and photon energies.
  • Typically, electromagnetic waves travel at the speed of light in a vacuum. 
  • The electromagnetic spectrum is as follows: 

​

EXPLANATION:

• The velocity of an electromagnetic wave (c) in a vacuum is related to its wavelength (λ) and frequency (f) as follows:

f λ = c 

• As we go from gamma rays to radio waves, the frequency decreases, and the wavelength increases.
• Hence, the velocity (c = 3 × 10⁸ m/s) is constant.

The correct option is 4.

CONCEPT:

Longitudinal wave motion: 

• It is that wave motion in which individual particles of the medium execute simple harmonic motion about their mean position along the same direction, in which the wave is propagated.
• For example Sound waves.

Transverse wave motion: 

• The vibration of particles in a medium and the propagation of waves are perpendicular to each other.
• Example: Waves in the surface of the water, waves in a string, Electromagnetic waves, etc.

Polarization: 

• This is a process by which unpolarised light is transformed into polarized light.
• Polarisation tells about the wave nature of light, as the light wave is polarised in a particular plane.
• The longitudinal waves cannot be polarised.

EXPLANATION:

• The sound wave is a longitudinal wave that can't be polarised. 
• The electromagnetic wave is a transverse wave and it can be polarised. 
• Thus the polarisation differentiates the electromagnetic waves and sound waves.

Concept:

In electromagnetic waves, the ratio of amplitudes of the electric field and the magnetic field is equal to the velocity of the electromagnetic waves in free space.

\(\frac{{{E_0}}}{{{B_0}}} = c\)

\(\frac{{{B_0}}}{{{E_0}}} = \frac{1}{c}\)

Where:

E₀ = Electric field

B₀ = Magnetic field

c = velocity

Derivation:

The intrinsic impedance of the wave is defined as the ratio of the electric field and the magnetic field phasor (complex amplitude), i.e.

\(\frac{{\left| E \right|}}{{\left| H \right|}} = \eta = \sqrt {\frac{\mu }{\epsilon}} \)

Since B = μ H

\(H = \frac{B}{\mu }\)

\(\frac{{\left| E \right|}}{{\left| B \right|}}\mu = \sqrt {\frac{\mu }{\epsilon}}\)

\(\frac{{\left| E \right|}}{{\left| B \right|}} = \sqrt {\frac{1}{{\mu\epsilon }}}\)

\(c = \frac{1}{{\sqrt {{\mu _0}{\epsilon_o}} }},\;\frac{{\left| E \right|}}{{\left| B \right|}} = c\)

CONCEPT:

• Light: It is an electromagnetic wave.
  • Light shows dual nature i.e. wave and particle.

Phenomenon explaining the wave nature of light is/are:

1. Interference
2. Diffraction
3. Polarisation

Phenomenons explaining the particle nature of light are:

1. Photoelectric effect
2. Compton Scattering

• Photons associated with light shows that the energy is Quantised and occurs in discrete levels.

EXPLANATION:

• From the above discussion, the light has a dual nature. So option 4 is correct.

CONCEPT:

• Electromagnetic waves: It is a transverse wave defined as composed of oscillating electrical and magnetic fields perpendicular (90°) to each other.
• The Speed of electromagnetic waves is the same as of speed of light c = 3 × 10⁸ m/s.

EXPLANATION:

• James Clerk Maxwell: He is a Scottish Mathematical physicist. He proposed equations of Magnetism (Maxwell’s equations of electromagnetism) and predicted that electromagnetic waves exist but didn’t discover it.
• Heinrich Rudolf Hertz: He is a German physicist who conclusively proved the existence of electromagnetic waves. The unit of frequency, Cycles per second was named Hertz in his honour.
• Michael Faraday: He is an English Scientist who discovered principles of Electromagnetic induction, Electrolysis and Diamagnetism. The SI Unit of capacitance is Farad in his honour
• Hans Christian Oersted: He was a Danish physicist who finds out that electric current creates a magnetic field.

So, Option 2 is correct.