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) \( 2(H/\alpha)^{1/2 \)}.

Answer 2

Because the air's refractive index increases with height above the road (\( \mu = (1+\alpha y)^{1/2} \)), a ray of light leaving a point on the road bends as it travels, curving upward before it reaches the man's eye — this is the same bending effect behind mirages. We can use the grazing-ray condition at the road surface to relate the visible distance to \( H \) and \( \alpha \), then compare with each option.

1. \( 2(H/\alpha)^{1/2} \): Applying the grazing-incidence condition \( \mu(0)\sin(90^\circ) = \mu(y)\sin\phi \) along the curved ray path and integrating the resulting trajectory from the road up to the man's eye height \( H \), the horizontal distance the man can see out to works out to \( 2\sqrt{H/\alpha} \), matching this option exactly.
2. \( 3(H/\alpha)^{1/2} \): This would arise from a different (steeper) growth assumption for \( \mu \) with height than the given \( (1+\alpha y)^{1/2} \) dependence, and does not match the integration of the actual ray path for this refractive-index profile.
3. \( (H/\alpha)^{1/2} \): This is smaller by exactly a factor of 2 compared with the properly integrated ray path result, missing the factor that comes out of the trajectory calculation.
4. \( (H\alpha)^{1/2} \): This has \( \alpha \) multiplying \( H \) rather than dividing it, which would mean a larger \( \alpha \) (faster increase of refractive index with height) lets the man see farther — the opposite of what should physically happen, since a faster-changing refractive index bends rays more sharply and should shorten, not lengthen, the visible distance.

Carrying the ray-bending calculation through for this specific \( \mu(y) \) profile, using the grazing condition at the road and integrating up to height \( H \), consistently produces the factor of 2 seen in this option.

Therefore, the correct answer is \( 2(H/\alpha)^{1/2} \).