Question

The magnetic field in a plane electromagnetic wave travelling in glass (\( n = 1.5 \)) is given by \[ B_y = (2 \times 10^{-7} \text{ T}) \sin(\alpha x + 1.5 \times 10^{11} t) \] where \( x \) is in metres and \( t \) is in seconds. The value of \( \alpha \) is:

• A. \( 0.5 \times 10^3 \, \text{m}^{-1} \)
• B. \( 6.0 \times 10^2 \, \text{m}^{-1} \)
• C. \( 7.5 \times 10^2 \, \text{m}^{-1} \)
• D. \( 1.5 \times 10^3 \, \text{m}^{-1} \)

Hint:

For EM waves in a medium: \( k = \frac{\omega}{v} \), and \( v = \frac{c}{n} \). Always find speed first, then wave number.

Answer 1

The Correct Option is C

Concept: A plane electromagnetic wave is represented as: \[ \sin(kx \pm \omega t) \] Where:

\( k \) = wave number \( = \frac{\omega}{v} \)
\( \omega \) = angular frequency
\( v \) = speed of wave in medium
Speed of EM wave in medium: \[ v = \frac{c}{n} \]
Step 1: Identify given quantities. From the wave equation: \[ \omega = 1.5 \times 10^{11} \, \text{rad/s} \] Refractive index: \[ n = 1.5 \]
Step 2: Speed of wave in glass. \[ v = \frac{c}{n} = \frac{3 \times 10^8}{1.5} = 2 \times 10^8 \, \text{m/s} \]
Step 3: Calculate wave number \( k \). \[ k = \frac{\omega}{v} \] \[ k = \frac{1.5 \times 10^{11}}{2 \times 10^8} = 0.75 \times 10^3 = 7.5 \times 10^2 \, \text{m}^{-1} \]
Step 4: Identify \( \alpha \). Comparing with \( \sin(\alpha x + \omega t) \), we get: \[ \alpha = k = 7.5 \times 10^2 \, \text{m}^{-1} \]