Question

An electromagnetic wave travels in free space along the \(x\)-direction. At a particular point in space and time, \[ \vec{B} = 2 \times 10^{-7}\,\hat{j}\,\text{T} \] is associated with this wave. The value of corresponding electric field \(\vec{E}\) at this point is __________ V/m.

• A. \(60\hat{k}\)
• B. \(-60\hat{k}\)
• C. \(30\hat{k}\)
• D. \(-600\hat{k}\)

Hint:

Remember for electromagnetic waves:

• \(E = cB\)
• \(\vec{E}\), \(\vec{B}\), and direction of propagation are mutually perpendicular
• Use right-hand rule: \[ \vec{E} \times \vec{B} = \text{Propagation direction} \]

Answer 1

The Correct Option is A

Concept: For an electromagnetic wave propagating in free space: \[ E = cB \] where:

• \(E\) = magnitude of electric field
• \(B\) = magnitude of magnetic field
• \(c = 3 \times 10^8 \,\text{m/s}\) (speed of light)

Also, the directions of \(\vec{E}\), \(\vec{B}\), and propagation are mutually perpendicular and follow: \[ \vec{E} \times \vec{B} = \text{Direction of propagation} \]

Step 1: Calculate the magnitude of electric field. Given: \[ B = 2 \times 10^{-7}\,\text{T} \] Using: \[ E = cB \] Substituting values: \[ E = (3 \times 10^8)(2 \times 10^{-7}) \] \[ E = 6 \times 10^1 \] \[ E = 60\,\text{V/m} \]

Step 2: Determine the direction of electric field. The wave propagates along: \[ +\hat{i} \] Given: \[ \vec{B} = 2 \times 10^{-7}\hat{j} \] Using: \[ \vec{E} \times \vec{B} = \hat{i} \] We know: \[ \hat{k} \times \hat{j} = -\hat{i} \] Therefore: \[ (-\hat{k}) \times \hat{j} = \hat{i} \] Hence, electric field is along: \[ -\hat{k} \] Therefore, \[ \boxed{\vec{E} = -60\,\hat{k}\,\text{V/m}} \]