Question

A plane EM wave is propagating along x direction. It has a wavelength of 4 mm. If electric field is in y direction with the maximum magnitude of 60 V m^-1, the equation for magnetic field is:

• A. \( B_z = 60 \sin \left[ \frac{\pi}{2} \left( x - 3 \times 10^8 t \right) \right] \hat{k} \, \text{T} \)
• B. \( B_z = 2 \times 10^{-7} \sin \left[ \frac{\pi}{2} \times 10^3 \left( x - 3 \times 10^8 t \right) \right] \hat{k} \, \text{T} \)
• C. \( B_x = 60 \sin \left[ \frac{\pi}{2} \left( x - 3 \times 10^8 t \right) \right] \hat{i} \, \text{T} \)
• D. \( B_z = 2 \times 10^{-7} \sin \left[ \frac{\pi}{2} \left( x - 3 \times 10^8 t \right) \right] \hat{k} \, \text{T} \)

Answer 1

The Correct Option is B

Step 1: Relation between electric and magnetic fields

The relationship between the electric field \(E\) and magnetic field \(B\) is:

\[ E = cB, \]

where \(c = 3 \times 10^8 \, \text{m/s}\).

Substitute \(E = 60 \, \text{Vm}^{-1}\):

\[ 60 = 3 \times 10^8 \cdot B \implies B = \frac{60}{3 \times 10^8} = 2 \times 10^{-7} \, \text{T}. \]

Step 2: Calculate the frequency

The wavelength is given as:

\[ \lambda = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m}. \]

The wave velocity \(c\) is related to the frequency \(f\) as:

\[ c = f\lambda \implies f = \frac{c}{\lambda} = \frac{3 \times 10^8}{4 \times 10^{-3}} = \frac{3}{4} \times 10^{11} \, \text{Hz}. \]

Step 3: Angular frequency

The angular frequency \(\omega\) is given by:

\[ \omega = 2\pi f = 2\pi \cdot \frac{3}{4} \times 10^{11} = \frac{3\pi}{2} \times 10^{11}. \]

Thus:

\[ \omega = \frac{\pi}{2} \times 10^3. \]

Step 4: Determine the direction of the fields

• The electric field is in the \(y\)-direction (\(\hat{\jmath}\)).
• The wave propagates in the \(x\)-direction (\(\hat{\imath}\)).
• The magnetic field must be perpendicular to both \(\hat{\imath}\) and \(\hat{\jmath}\), i.e., in the \(z\)-direction (\(\hat{k}\)).

Step 5: Equation of the magnetic field

The magnetic field \(B_z\) is:

\[ B_z = 2 \times 10^{-7} \sin\left[\frac{\pi}{2} \times 10^3 \left(x - 3 \times 10^8 t\right)\right] \hat{k}. \]

Final Answer: \[ B_z = 2 \times 10^{-7} \sin\left[\frac{\pi}{2} \times 10^3 \left(x - 3 \times 10^8 t\right)\right] \, \hat{k} \, \text{kT}. \]