Question

Equation of an electromagnetic wave in a medium is given by \[ E = 2\sin\left(2\times 10^{15}t - 10^{7}x\right). \] Find the refractive index of the medium.

• A. \(\dfrac{3}{2}\)
• B. \(2\)
• C. \(\dfrac{5}{3}\)
• D. \(\dfrac{4}{3}\)

Hint:

For EM wave equations:

Identify \(\omega\) and \(k\) directly from the equation.
Use \(v = \dfrac{\omega}{k}\).
Refractive index is always \(n = \dfrac{c}{v}\).

Answer 1

The Correct Option is A

Concept: The general equation of a plane electromagnetic wave is: \[ E = E_0 \sin(\omega t - kx) \] where:

\(\omega\) is the angular frequency,
\(k\) is the wave number,
Wave speed \(v = \dfrac{\omega}{k}\),
Refractive index \(n = \dfrac{c}{v}\).
Step 1: Compare the given equation with the standard form. Given: \[ E = 2\sin\left(2\times 10^{15}t - 10^{7}x\right) \] Hence, \[ \omega = 2\times 10^{15}\ \text{rad s}^{-1}, \quad k = 10^{7}\ \text{m}^{-1} \]
Step 2: Calculate the velocity of the wave in the medium. \[ v = \frac{\omega}{k} = \frac{2\times 10^{15}}{10^{7}} = 2\times 10^{8}\ \text{m s}^{-1} \]
Step 3: Calculate the refractive index. Speed of light in vacuum: \[ c = 3\times 10^{8}\ \text{m s}^{-1} \] \[ n = \frac{c}{v} = \frac{3\times 10^{8}}{2\times 10^{8}} = \frac{3}{2} \]
Conclusion: \[ \boxed{n = \dfrac{3}{2}} \] Hence, the correct answer is (1).