Question

A man of height \( H \) is standing on level road where because of temperature variation, the refractive index of air is varying as \( \mu = (1 + \alpha y)^{1/2 \), where \( y \) is the height from the level road and \( \alpha \) is a positive constant. Find the distant point that man can see on the road.}

• A. \( 2(H/\alpha)^{1/2} \)
• B. \( 3(H/\alpha)^{1/2} \)
• C. \( (H/\alpha)^{1/2} \)
• D. \( (H\alpha)^{1/2} \)

Hint:

In problems involving temperature variation and refractive index, the bending of light is influenced by the refractive index gradient.

Answer 1

The Correct Option is A

Step 1: Understanding the variation of refractive index.
The refractive index is given as \( \mu = (1 + \alpha y)^{1/2} \), where \( y \) is the height. The ray of light will be bent due to this variation in the refractive index. The path of the light ray will curve towards the road. The angle of deviation increases with height. Step 2: Calculating the distance.
From the formula for the refraction index and using the concept of ray bending, the distance that the man can see on the road depends on the height \( H \) and the parameter \( \alpha \). Solving the refraction problem gives the distance as \( 2(H/\alpha)^{1/2} \). Step 3: Conclusion.
The correct answer is (1) \( 2(H/\alpha)^{1/2 \)}.