Question

An EM wave propagating in x-direction has a wavelength of 8 mm. The electric field vibrating y-direction has maximum magnitude of 60 Vm^–1. Choose the correct equations for electric and magnetic field if the EM wave is propagating in vacuum:

• A. \(E_y=60\sin⁡[\frac{π}{4}×10^3(x−3×10^8t)]\hat{j}Vm^{−1} \)
  \(B_y = 2\sin\left[\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right]\hat{k}T\)
• B. \(E_y = 60\sin\left[\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right] \hat{j} \,Vm^{-1}\) 
  \(B_z = 2 \times 10^{-7} \sin\left[\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right] \hat{k} \, \text{T}\)
• C. \(E_y = 2 \times 10^{-7} \sin\left(\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right) \hat{j} \, Vm^{-1}\)
  \(B_z = 60 \sin\left[\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right]\hat{k} \, \text{T}\)
• D. \(E_y = 2 \times 10^{-7} \sin\left[\frac{\pi}{4} \times 10^4 (x - 4 \times 10^8 t)\right] \hat{j} \, Vm^{-1}\)
  \(B_z = 60 \sin\left[\frac{\pi}{4} \times 10^4 x - 4 \times 10^8 t)\right] \hat{k} \, \text{T}\)

Answer 1

The Correct Option is B

To solve the given problem, let's start by reviewing the fundamental concepts of electromagnetic (EM) waves and their properties:

• Electromagnetic waves are characterized by oscillating electric and magnetic fields, which are perpendicular to each other and to the direction of wave propagation. 
• The general form of an electromagnetic wave propagating in the x-direction is given by the equations: \(E = E_0 \sin(kx - \omega t) \hat{j}\) for the electric field, and \(B = B_0 \sin(kx - \omega t) \hat{k}\) for the magnetic field.
• In a vacuum, the speed of electromagnetic waves, \( c \), is approximately \( 3 \times 10^8 \, \text{m/s} \), which relates the wave's frequency and wavelength.

Given data in the problem:

• Wavelength \((\lambda) = 8 \, \text{mm} = 8 \times 10^{-3} \, \text{m}\)
• Maximum electric field magnitude \((E_0) = 60 \, \text{Vm}^{-1}\)
• The wave is propagating in a vacuum along the x-direction.

First, let's find the wave number \( k \) and the angular frequency \( \omega \):

• Wave number, \( k \): It is given by \( k = \frac{2\pi}{\lambda} \).
  • Substitute the given wavelength: \(k = \frac{2\pi}{8 \times 10^{-3}} = \frac{\pi}{4} \times 10^{3} \, \text{m}^{-1}\)
• Angular frequency, \( \omega \): As the wave is in a vacuum, \( \omega = ck \).
  • Substitute the speed of light: \(\omega = (3 \times 10^8) \times \left(\frac{\pi}{4} \times 10^{3}\right) = \frac{3\pi}{4} \times 10^{11} \, \text{rad/s}\)

Given options, the correct equations for such a wave are chosen under the following considerations:

• Amplitude of electric field \((E_0 = 60 \, \text{Vm}^{-1})\)
• The relationship between electric and magnetic fields in a vacuum is \(B_0 = \frac{E_0}{c}\).

Calculate \( B_0 \): \(B_0 = \frac{60}{3 \times 10^8} = 2 \times 10^{-7} \, \text{T}\)

Therefore, the correct option is:

\(E_y = 60\sin\left[\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right] \hat{j} \,Vm^{-1}\) 
\(B_z = 2 \times 10^{-7} \sin\left[\frac{\pi}{4} \times 10^3 (x - 3 \times 10^8 t)\right] \hat{k} \, \text{T}\)