Question

A capillary tube of radius \( r \) is dipped in water; find the height \( h \) to which water rises if the surface tension is \( T \).

• A. \( \dfrac{T}{\rho g r} \)
• B. \( \dfrac{2T}{\rho g r} \)
• C. \( \dfrac{4T}{\rho g r} \)
• D. \( \dfrac{T}{2\rho g r} \)

Hint:

Capillary rise is inversely proportional to the radius of the tube. Narrower tubes cause higher rise of liquid.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the formula for capillary rise, which describes how much a liquid ascends in a narrow tube due to surface tension.
Step 2: Key Formula or Approach:
The ascent formula for capillary rise is:
\[ h = \frac{2T \cos \theta}{r \rho g} \]
where \( T \) is surface tension, \( \theta \) is the angle of contact, \( \rho \) is density, and \( g \) is gravity.
Step 3: Detailed Explanation:
For water in contact with glass, the angle of contact \( \theta \) is approximately \( 0^\circ \).
Substituting \( \cos 0^\circ = 1 \) into the formula:
\[ h = \frac{2T(1)}{r \rho g} \]
\[ h = \frac{2T}{\rho g r} \]
This equation represents the balance between the vertical component of the surface tension force and the weight of the liquid column.
Step 4: Final Answer:
The height of the water rise is \( \frac{2T}{\rho g r} \).