Question

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

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

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In an electromagnetic wave, the electric field \(\vec{E}\), magnetic field \(\vec{B}\), and the direction of propagation \(\vec{v}\) are all mutually perpendicular.
Their relationship is given by the cross product \(\hat{E} \times \hat{B} = \hat{v}\).
The magnitude of the electric field is tied to the magnetic field by the speed of light \(c\).
Step 2: Key Formula or Approach:
Magnitude relationship: \(E = cB\), where \(c = 3 \times 10^8 \text{ m/s}\).
Directional relationship: The direction of wave propagation is given by \(\vec{E} \times \vec{B}\).
Step 3: Detailed Explanation:
First, calculate the magnitude of the electric field:
\[ E = (3 \times 10^8 \text{ m/s}) \times (2 \times 10^{-7} \text{ T}) = 60 \text{ V/m} \] Next, determine the direction of the electric field.
The wave travels along the \(x\)-direction, so the propagation vector is \(+\hat{i}\).
The magnetic field is along the \(y\)-direction, so \(\hat{B} = \hat{j}\).
We need to find the unit vector \(\hat{E}\) such that \(\hat{E} \times \hat{j} = \hat{i}\).
Recall the standard cross product rules for Cartesian unit vectors: \(\hat{k} \times \hat{j} = -\hat{i}\).
Therefore, \((-\hat{k}) \times \hat{j} = \hat{i}\).
This means the electric field must be directed along the negative \(z\)-axis, so \(\hat{E} = -\hat{k}\).
Combining magnitude and direction:
\[ \vec{E} = -60\hat{k} \text{ V/m} \] Step 4: Final Answer:
The corresponding electric field is \(-60\hat{k}\).