Question

The electric field of an electromagnetic wave travelling through a medium is given by \[ \vec{E}(x,t)=25\sin(2\times10^{15}t-10^{7}x)\,\hat{n}. \] Then the refractive index of the medium is ________. (All given measurements are in SI units)

• A. \(1.7\)
• B. \(1.5\)
• C. \(1.2\)
• D. \(2\)

Hint:

Always compare the given wave equation with \( \sin(\omega t-kx) \) to directly read \( \omega \) and \( k \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The speed of light in a medium is determined by the wave parameters (angular frequency \(\omega\) and wave number \(k\)). The refractive index is the ratio of the vacuum speed of light to the medium speed.
Step 2: Key Formula or Approach:
Standard wave equation: \(E = E_0 \sin(\omega t - kx)\).
Speed of wave in medium: \(v = \frac{\omega}{k}\).
Refractive index: \(n = \frac{c}{v}\).
Step 3: Detailed Explanation:
From the given equation:
\(\omega = 2.0 \times 10^{15} \text{ rad/s}\)
\(k = 10^7 \text{ m}^{-1}\)
Velocity in the medium:
\[ v = \frac{\omega}{k} = \frac{2.0 \times 10^{15}}{10^7} = 2.0 \times 10^8 \text{ m/s} \]
We know the speed of light in vacuum is \(c = 3.0 \times 10^8 \text{ m/s}\).
Refractive index:
\[ n = \frac{c}{v} = \frac{3 \times 10^8}{2 \times 10^8} = 1.5 \]
Step 4: Final Answer:
The refractive index of the medium is \(1.5\).