Question

A plane electromagnetic wave of frequency 20 MHz travels in free space along the +x direction. At a particular point in space and time, the electric field vector of the wave is \( E_y = 9.3 \, \text{V/m} \). Then, the magnetic field vector of the wave at that point is:

• A. \( B_z = 9.3 \times 10^{-8} \, \text{T} \)
• B. \( B_z = 1.55 \times 10^{-8} \, \text{T} \)
• C. \( B_z = 6.2 \times 10^{-8} \, \text{T} \)
• D. \( B_z = 3.1 \times 10^{-8} \, \text{T} \)

Hint:

The magnetic field of an electromagnetic wave is related to the electric field by \( B = \frac{E}{c} \), where \( c \) is the speed of light.

Answer 1

The Correct Option is D

To solve this problem, we need to find the magnetic field vector \( B_z \) of a plane electromagnetic wave given its electric field vector \( E_y \) and the frequency of the wave. 

First, recall that in electromagnetic waves traveling in free space, the electric field \( \mathbf{E} \) and magnetic field \( \mathbf{B} \) are related by the following relation:

\(E = cB\)

where:

• \(E = 9.3 \, \text{V/m}\) (Given electric field amplitude)
• \(c = 3 \times 10^8 \, \text{m/s}\) (Speed of light in free space)
• \(B\) is the magnetic field amplitude we need to find.

Step 1: Use the relation \(E = cB\), we solve for \(B\):

\(B = \frac{E}{c}\)

Step 2: Substitute the given values:

\(B = \frac{9.3}{3 \times 10^8}\)

Step 3: Calculate the magnetic field amplitude:

\(B = 3.1 \times 10^{-8} \, \text{T}\)

Therefore, the magnetic field vector of the wave at that point is:

Correct Answer: \(B_z = 3.1 \times 10^{-8} \, \text{T}\)

This matches option 4, confirming our calculation is correct. The problem uses the direct relationship between the electric and magnetic fields in electromagnetic waves, which is a fundamental principle of wave propagation in physics.

Answer 2

For a plane electromagnetic wave, the relationship between the electric field \( E \) and the magnetic field \( B \) is given by: \[ B = \frac{E}{c}, \] where \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light in a vacuum. Given: \[ E = 9.3 \, \text{V/m}, \quad f = 20 \, \text{MHz}. \] The wavelength \( \lambda \) of the wave can be found using the relationship: \[ c = f \lambda \quad \Rightarrow \quad \lambda = \frac{c}{f}. \] Now, using the value \( E = 9.3 \, \text{V/m} \), the magnetic field \( B \) is: \[ B = \frac{9.3}{3 \times 10^8} = 3.1 \times 10^{-8} \, \text{T}. \]