Question

A plane electromagnetic wave travels through a medium and the magnetic field associated with it is given by
\( B = 5 \times 10^{-8} \sin (3 \times 10^{10} t - 150 x) \) T
where x is in metres and t is in seconds.
The velocity of the wave is :

• A. \( 2.0 \times 10^8 \text{ ms}^{-1} \)
• B. \( 4.5 \times 10^7 \text{ ms}^{-1} \)
• C. \( 3.5 \times 10^7 \text{ ms}^{-1} \)
• D. \( 2.5 \times 10^8 \text{ ms}^{-1} \)

Hint:

Always remember that wave speed is simply the coefficient of \( t \) divided by the coefficient of \( x \), regardless of the order they appear inside the sine or cosine function.

Answer 1

The Correct Option is A

Concept:
• A plane electromagnetic wave's electric and magnetic fields oscillate sinusoidally.

• The standard mathematical representation for a wave traveling in the positive x-direction is \( B = B_0 \sin(\omega t - kx) \) or \( B = B_0 \sin(kx - \omega t) \).

• Here, \( \omega \) represents the angular frequency in rad/s, and \( k \) represents the wave number in rad/m.

• The propagation speed (velocity) \( v \) of the wave is related to these parameters by the formula \( v = \frac{\omega}{k} \).

Step 1: Extract parameters from the given wave equation
The given equation for the magnetic field is:
\[ B = 5 \times 10^{-8} \sin (3 \times 10^{10} t - 150 x) \text{ T} \]
By comparing this with the standard form \( B = B_0 \sin(\omega t - kx) \), we can identify:
The angular frequency \( \omega \) is the coefficient of \( t \), so \( \omega = 3 \times 10^{10} \text{ rad/s} \).
The wave number \( k \) is the coefficient of \( x \), so \( k = 150 \text{ rad/m} \).

Step 2: Calculate the velocity of the wave
Substitute the extracted values of \( \omega \) and \( k \) into the wave velocity formula:
\[ v = \frac{\omega}{k} \]
\[ v = \frac{3 \times 10^{10}}{150} \]
To simplify, break down the numerator:
\[ v = \frac{300 \times 10^8}{150} \]
\[ v = 2.0 \times 10^8 \text{ m/s} \]

Step 3: Conclusion
The calculated velocity of the electromagnetic wave is \( 2.0 \times 10^8 \text{ ms}^{-1} \).
This result matches option (A).