Question

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

• A. \(2\cdot0 \times 10^8\text{ ms}^{-1}\)
• B. \(4\cdot5 \times 10^7\text{ ms}^{-1}\)
• C. \(3\cdot5 \times 10^7\text{ ms}^{-1}\)
• D. \(2\cdot5 \times 10^8\text{ ms}^{-1}\)

Hint:

For any wave equation expressed in terms of \(( \omega t - kx )\), the wave speed is always equal to the coefficient of \(t\) divided by the coefficient of \(x\). This rule allows you to quickly solve wave speed problems without needing to memorize secondary relations.

Answer 1

The Correct Option is A

Concept: A standard sinusoidal traveling wave moving along the positive \(x\)-axis can be mathematically described by the wave equation of the form: \[ B(x, t) = B_0 \sin(\omega t - kx) \] where:
• \(B_0\) is the maximum amplitude of the magnetic field.
• \(\omega\) is the angular frequency of the wave, related to the frequency \(f\) by \(\omega = 2\pi f\).
• \(k\) is the wave number (propagation constant), related to the wavelength \(\lambda\) by \(k = \frac{2\pi}{\lambda}\). The propagation speed (velocity \(v\)) of any such wave is uniquely determined by the ratio of its angular frequency to its wave number: \[ v = \frac{\omega}{k} \]

Step 1: Extrapolating the values of \(\omega\) and \(k\) by comparing equations.
The given equation for the magnetic field of the electromagnetic wave is: \[ B = 5 \times 10^{-8} \sin (3 \times 10^{10} t - 150 x) \] By comparing this directly with the standard wave equation \(B = B_0 \sin(\omega t - kx)\), we can identify the corresponding coefficients:
• The coefficient of \(t\) is the angular frequency: \(\omega = 3 \times 10^{10}\text{ rad/s}\)
• The coefficient of \(x\) is the wave number: \(k = 150\text{ m}^{-1}\)

Step 2: Computing the wave velocity.
Using the velocity formula for electromagnetic waves in a medium: \[ v = \frac{\omega}{k} \] Substitute the values extracted in Step 1 into this equation: \[ v = \frac{3 \times 10^{10}}{150} \] To simplify the division, we can rewrite \(3 \times 10^{10}\) as \(30 \times 10^9\): \[ v = \frac{30 \times 10^9}{150} = \frac{3 \times 10^9}{15} = \frac{1}{5} \times 10^9 \] Converting the fraction into decimal form: \[ v = 0.2 \times 10^9\text{ ms}^{-1} \] Expressing this in standard scientific notation gives: \[ v = 2.0 \times 10^8\text{ ms}^{-1} \] Thus, the velocity of the electromagnetic wave through the given medium is \(2.0 \times 10^8\text{ ms}^{-1}\), which corresponds to Option (A).