Question:
An electromagnetic wave, whose wave normal makes an angle of \( 45^\circ \) with the vertical, is travelling in air and strikes a horizontal liquid surface. While travelling through the liquid, it gets deviated by \( 15^\circ \). If the speed of electromagnetic wave in air is \( 3\times10^8 \text{ m/s} \), then the speed of electromagnetic wave in the liquid will be
An electromagnetic wave, whose wave normal makes an angle of \( 45^\circ \) with the vertical, is travelling in air and strikes a horizontal liquid surface. While travelling through the liquid, it gets deviated by \( 15^\circ \). If the speed of electromagnetic wave in air is \( 3\times10^8 \text{ m/s} \), then the speed of electromagnetic wave in the liquid will be
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Remember that \( \frac{1}{\sqrt{2}} \approx 0.707 \). To quickly compute \( \frac{3}{\sqrt{2}} \), you can multiply \( 3 \times 0.707 = 2.12 \). Performing such simple mental calculations helps to find the correct decimal option instantly without extensive division.
Updated On: May 28, 2026
- \( \frac{\sqrt{2}}{3}\times10^8 \text{ m/s} \)
- \( 1.5\times10^8 \text{ m/s} \)
- \( 2.1\times10^8 \text{ m/s} \)
- \( 2.5\times10^8 \text{ m/s} \)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
An electromagnetic wave travels from air to a liquid medium, bending towards the normal. We are given the angle of incidence \( i \), the angle of deviation \( \delta \), and the speed of light in air. We need to find the speed of the wave in the liquid.
Step 2: Key Formula or Approach:
1. Snell's Law:
\[ \mu_{\text{air}} \sin i = \mu_{\text{liquid}} \sin r \]
2. Angle of Refraction (\( r \)): Since the light travels from a rarer to a denser medium, it bends towards the normal, so:
\[ r = i - \delta \]
3. Speed in the medium:
\[ v = \frac{c}{\mu_{\text{liquid}}} \]
Step 3: Detailed Explanation:
1. Identify the angles:
- Angle of incidence, \( i = 45^\circ \)
- Deviation, \( \delta = 15^\circ \)
- Angle of refraction:
\[ r = i - \delta = 45^\circ - 15^\circ = 30^\circ \]
2. Calculate the refractive index of the liquid \( \mu_{\text{liquid}} \) using Snell's Law (taking \( \mu_{\text{air}} = 1 \)):
\[ 1 \cdot \sin(45^\circ) = \mu_{\text{liquid}} \sin(30^\circ) \]
\[ \frac{1}{\sqrt{2}} = \mu_{\text{liquid}} \left(\frac{1}{2}\right) \]
\[ \mu_{\text{liquid}} = \sqrt{2} \approx 1.414 \]
3. Calculate the speed of light \( v \) in the liquid:
\[ v = \frac{c}{\mu_{\text{liquid}}} = \frac{3 \times 10^8 \text{ m/s}}{\sqrt{2}} \]
\[ v = \frac{3}{1.414} \times 10^8 \text{ m/s} \approx 2.12 \times 10^8 \text{ m/s} \]
This value is closest to \( 2.1 \times 10^8 \text{ m/s} \).
Step 4: Final Answer:
The speed of the electromagnetic wave in the liquid is \( 2.1\times10^8 \text{ m/s} \).
An electromagnetic wave travels from air to a liquid medium, bending towards the normal. We are given the angle of incidence \( i \), the angle of deviation \( \delta \), and the speed of light in air. We need to find the speed of the wave in the liquid.
Step 2: Key Formula or Approach:
1. Snell's Law:
\[ \mu_{\text{air}} \sin i = \mu_{\text{liquid}} \sin r \]
2. Angle of Refraction (\( r \)): Since the light travels from a rarer to a denser medium, it bends towards the normal, so:
\[ r = i - \delta \]
3. Speed in the medium:
\[ v = \frac{c}{\mu_{\text{liquid}}} \]
Step 3: Detailed Explanation:
1. Identify the angles:
- Angle of incidence, \( i = 45^\circ \)
- Deviation, \( \delta = 15^\circ \)
- Angle of refraction:
\[ r = i - \delta = 45^\circ - 15^\circ = 30^\circ \]
2. Calculate the refractive index of the liquid \( \mu_{\text{liquid}} \) using Snell's Law (taking \( \mu_{\text{air}} = 1 \)):
\[ 1 \cdot \sin(45^\circ) = \mu_{\text{liquid}} \sin(30^\circ) \]
\[ \frac{1}{\sqrt{2}} = \mu_{\text{liquid}} \left(\frac{1}{2}\right) \]
\[ \mu_{\text{liquid}} = \sqrt{2} \approx 1.414 \]
3. Calculate the speed of light \( v \) in the liquid:
\[ v = \frac{c}{\mu_{\text{liquid}}} = \frac{3 \times 10^8 \text{ m/s}}{\sqrt{2}} \]
\[ v = \frac{3}{1.414} \times 10^8 \text{ m/s} \approx 2.12 \times 10^8 \text{ m/s} \]
This value is closest to \( 2.1 \times 10^8 \text{ m/s} \).
Step 4: Final Answer:
The speed of the electromagnetic wave in the liquid is \( 2.1\times10^8 \text{ m/s} \).
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