Question

Electric field in a plane electromagnetic wave is given by E = 50 sin(500x - 10 \(\times\) 10\(^{10}\)t) V/m. The velocity of electromagnetic wave in this medium is : (Given C = speed of light in vacuum)

• A. \(\frac{2}{3}\)C
• B. C
• C. \(\frac{3}{2}\)C
• D. \(\frac{C}{2}\)

Hint:

Whenever you see a wave equation in the form \(\sin(ax \pm bt)\) or \(\cos(ax \pm bt)\), the wave speed is simply the ratio of the coefficient of time to the coefficient of position: \(v = |b/a|\). This is a quick way to find the speed without remembering the names 'angular frequency' and 'wave number'.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the equation of the electric field of a plane electromagnetic wave propagating in a medium. We need to find the velocity of this wave.
Step 2: Key Formula or Approach:
The standard equation for a plane wave traveling in the positive x-direction is: \[ E = E_0 \sin(kx - \omega t) \] where: - \(k\) is the angular wave number (\(k = 2\pi/\lambda\)) - \(\omega\) is the angular frequency (\(\omega = 2\pi f\)) The velocity of the wave (\(v\)) is given by the ratio of the angular frequency to the angular wave number: \[ v = \frac{\omega}{k} \] Step 3: Detailed Explanation:
The given equation is: \[ E = 50 \sin(500x - 10 \times 10^{10}t) \] By comparing this with the standard wave equation \(E = E_0 \sin(kx - \omega t)\), we can identify the values of \(k\) and \(\omega\).
Angular wave number, \(k = 500\) rad/m.
Angular frequency, \(\omega = 10 \times 10^{10} = 1 \times 10^{11}\) rad/s.
Now, we can calculate the velocity of the wave in the medium: \[ v = \frac{\omega}{k} = \frac{1 \times 10^{11}}{500} = \frac{100 \times 10^9}{500} = \frac{1}{5} \times 10^9 = 0.2 \times 10^9 \text{ m/s} \] \[ v = 2 \times 10^8 \text{ m/s} \] The question asks for the velocity in terms of C, the speed of light in vacuum, where \(C = 3 \times 10^8\) m/s.
Let's find the ratio \(\frac{v}{C}\): \[ \frac{v}{C} = \frac{2 \times 10^8 \text{ m/s}}{3 \times 10^8 \text{ m/s}} = \frac{2}{3} \] Step 4: Final Answer:
The velocity of the electromagnetic wave in this medium is \(v = \frac{2}{3}C\).