Question

A plane electromagnetic wave of frequency 100 MHz is travelling in vacuum along the x-direction. At a particular point in space and time, \( \mathbf{B} = 2.0 \times 10^{-8}\,\hat{k}\,\text{T} \). What is \( \mathbf{E} \) at this point? ( \( c = 3 \times 10^8 \,\text{m/s} \) )

• A. 0.6 ĵ V/m
• B. 6.0 ĵ V/m
• C. 6.0 k̂ V/m
• D. 0.6 k̂ V/m

Hint:

Always use the cyclic rule: $\hat{i} \to \hat{j} \to \hat{k} \to \hat{i}$. Since velocity is $\hat{i}$, $\vec{E}$ and $\vec{B}$ must be along $\hat{j}$ and $\hat{k}$.

Answer 1

The Correct Option is B

Step 1: Magnitude: $E = Bc = (2.0 \times 10^{-8}) \times (3 \times 10^8) = 6.0 \text{ V/m}$.
Step 2: Direction: In an EM wave, $\vec{E}$, $\vec{B}$, and the direction of propagation ($\vec{v}$) are mutually perpendicular, satisfying $\hat{E} \times \hat{B} = \hat{v}$.
Step 3: $\hat{v} = \hat{i}$ and $\hat{B} = \hat{k}$. $\hat{E} \times \hat{k} = \hat{i} \implies \hat{E} = \hat{j}$ (Since $\hat{j} \times \hat{k} = \hat{i}$).
Step 4: $\vec{E} = 6.0 \hat{j} \text{ V/m}$.