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: Understanding the Concept
In a capillary tube, the height of the liquid column is given by Jurin's law.

Step 2: Key Formula or Approach
\[ h=\frac{2T\cos\theta}{r\rho g} \]
For the same liquid (water) in the same place, \(T\), \(\theta\), \(\rho\) and \(g\) are constant.

Step 3: Reduce
\[ h\propto\frac1r\ \Rightarrow\ h\,r=\frac{2T\cos\theta}{\rho g}=\text{constant} \]

Step 4: Check the options
\(h/r\) and \(h/r^2\) would mean a rise that grows with radius, which is opposite to what is seen. \(hr^2\) is wrong because the radius enters only once. So \(h_nr_n=\) constant, option (D).

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