Question

The electric field in a plane electromagnetic wave is given by : \( E_y = 69 \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \) V/m. The expression for magnetic field associated with this electromagnetic wave is_________ T.

• A. \( B_z = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \)
• B. \( B_y = 69 \sin[0.6 \times 10^3 x + 1.8 \times 10^{11} t] \)
• C. \( B_z = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x + 1.8 \times 10^{11} t] \)
• D. \( B_y = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \)

Hint:

To quickly find the direction of B, use the right-hand rule or the cyclic relation \(\hat{i} \times \hat{j} = \hat{k}\). Here, propagation is +x (\(\hat{i}\)) and E-field is +y (\(\hat{j}\)). So B must be +z (\(\hat{k}\)). For the amplitude, simply divide \(E_0\) by \(3 \times 10^8\). The sinusoidal part is always the same for both E and B.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the equation for the electric field of a plane electromagnetic (EM) wave and asked to find the corresponding equation for the magnetic field. This involves finding the amplitude, direction, and phase of the magnetic field.
Step 2: Key Formula or Approach:
For an EM wave in vacuum, the amplitudes of the electric field (\(E_0\)) and magnetic field (\(B_0\)) are related by the speed of light, c:
\[ B_0 = \frac{E_0}{c} \] where \( c = 3 \times 10^8 \) m/s.
The direction of the electric field (\(\vec{E}\)), magnetic field (\(\vec{B}\)), and propagation (\(\vec{k}\)) are mutually perpendicular, following the relation \(\vec{E} \times \vec{B}\) points in the direction of \(\vec{k}\). The magnetic field and electric field are in phase.
Step 3: Detailed Explanation:
1. Find the amplitude of the magnetic field (\(B_0\)):
From the given equation, the amplitude of the electric field is \(E_0 = 69\) V/m.
\[ B_0 = \frac{E_0}{c} = \frac{69}{3 \times 10^8} = 23 \times 10^{-8} \, \text{T} = 2.3 \times 10^{-7} \, \text{T} \] 2. Determine the direction of the magnetic field:
The electric field \(E_y\) oscillates along the y-axis (\(\hat{j}\)).
The wave propagation direction is determined by the term \(kx - \omega t\). A positive `x` and negative `t` term indicates propagation in the +x direction (\(\hat{i}\)).
The relation for direction is \(\vec{k} \propto \vec{E} \times \vec{B}\).
So, \(\hat{i} \propto \hat{j} \times \vec{B}\). For this to be true, the magnetic field \(\vec{B}\) must be in the +z direction (\(\hat{k}\)). (\(\hat{j} \times \hat{k} = \hat{i}\)).
Thus, the magnetic field component is \(B_z\).
3. Write the full expression:
The magnetic field is in phase with the electric field, so it has the same sinusoidal part.
\[ B_z = B_0 \sin[kx - \omega t] \] \[ B_z = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \, \text{T} \] Step 4: Final Answer:
The expression for the magnetic field is \( B_z = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \). This matches option (A).