Question

If we dip capillary tubes of different radii \(r_n\) in water and the water rises to different heights \(h_n\) in them, then (\(n = 1,2,3\) .........)

• A. \(\frac{h_n}{r_n^2} = \text{constant}\)
• B. \(\frac{h_n}{r_n} = \text{constant}\)
• C. \(h_nr_n^2 = \text{constant}\)
• D. \(h_nr_n = \text{constant}\)

Hint:

Jurin law: h = 2T cos(theta) / (r rho g), so h is inversely proportional to r.

Answer 1

The Correct Option is D

Step 1: Approach
Balance the weight of the water column against the surface tension force.

Step 2: Balance
Upward force $=2\pi rT\cos\theta$. Weight $=\pi r^2h\rho g$.

Step 3: Result
Setting them equal gives $h=\dfrac{2T\cos\theta}{r\rho g}$, so the product $hr=\dfrac{2T\cos\theta}{\rho g}$ is the same for every tube. Option (D).

Final Answer:
The product of rise and radius is constant, option (D). \[ \boxed{h_nr_n=\text{constant}} \]