Question

A capillary tube is vertically immersed in water, water rises up to a height \( h_1 \). When the whole arrangement is taken to a depth \( d \) in a mine, the water level rises up to height \( h_2 \). The ratio \( \frac{h_1}{h_2} \) is (R = radius of earth)

• A. \( 1 + \frac{2d}{R} \)
• B. \( 1 - \frac{d}{R} \)
• C. \( 1 + \frac{d}{R} \)
• D. \( 1 - \frac{2d}{R} \)

Hint:

At greater depths, the pressure increases, reducing the capillary rise. The relationship is inversely proportional to the depth and radius of the Earth.

Answer 1

The Correct Option is B

Step 1: Understanding the effect of depth on the capillary rise.
The capillary rise in a tube depends on the surface tension, which is influenced by the depth of the tube below the surface of the water. As the tube is taken to a depth \( d \) inside the mine, the effective height of water rise changes due to the increase in atmospheric pressure with depth. Step 2: Deriving the relation.
At a depth \( d \), the water level rises less due to the increased pressure, which is directly proportional to the height of the water. Therefore, the ratio of the heights at the surface and at depth \( d \) is: \[ \frac{h_1}{h_2} = 1 - \frac{d}{R} \] Step 3: Conclusion.
Thus, the correct answer is (B) \( 1 - \frac{d}{R} \).