Question

When a capillary is dipped vertically in water, rise of water in capillary is \( h \). The angle of contact is zero. Now the tube is depressed so that its length above the water surface is \( h/2 \). The new apparent angle of contact is

• A. \( \sin^{-1} (0.5) \)
• B. \( \sin^{-1} (0.7) \)
• C. \( \cos^{-1} (0.5) \)
• D. \( \cos^{-1} (0.7) \)

Hint:

The apparent contact angle in a capillary depends on the length of the tube above the liquid surface. When the tube is depressed, it affects the contact angle and the rise of liquid.

Answer 1

The Correct Option is C

Step 1: Understanding the rise of water.
In a capillary tube, the rise of liquid is given by: \[ h = \frac{2\gamma \cos \theta}{\rho g r} \] where \( \gamma \) is the surface tension, \( \theta \) is the angle of contact, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the capillary tube.
Step 2: Effect of depression on contact angle.
When the tube is depressed, the apparent contact angle is affected by the length above the water surface. The new apparent angle of contact is given by \( \cos^{-1} (0.5) \), which corresponds to the change in the capillary rise due to the depression.
Step 3: Conclusion.
Thus, the new apparent angle of contact is \( \cos^{-1} (0.5) \), corresponding to option (C).