Question

The lower end of a capillary tube is dipped into water and it is seen that water rises through \(7.5\,\text{cm}\) in the capillary. Given, surface tension of water is \(7.5 \times 10^{-2}\,\text{N m}^{-1}\) and angle of contact is zero. Find the diameter (in mm) of the capillary tube. (Given \(g = 10\,\text{m s}^{-2}\))

Hint:

Use \(h = \frac{2T}{\rho g r}\) for water (since \(\theta = 0^\circ\)).
Always convert cm to m carefully — most mistakes happen here!

Answer 1

Correct Answer: 0.4

Step 1: Understanding the Concept:
Capillary rise occurs due to surface tension. The height of the liquid column is determined by the balance between the upward force of surface tension and the downward force of gravity on the liquid column.
Step 2: Key Formula or Approach:
The height of capillary rise \(h\) is given by:
\[ h = \frac{2T \cos \theta}{r \rho g} \]
Where \(T\) is surface tension, \(\theta\) is the angle of contact, \(r\) is the radius, \(\rho\) is the density of the liquid, and \(g\) is gravity.
Step 3: Detailed Explanation:
Given:
\(h = 7.5\text{ cm} = 0.075\text{ m}\).
\(T = 7.5 \times 10^{-2}\text{ Nm}^{-1}\).
\(\theta = 0^{\circ} \implies \cos \theta = 1\).
\(\rho_{\text{water}} = 1000\text{ kg/m}^{3}\).
\(g = 10\text{ ms}^{-2}\).
Calculate the radius \(r\):
\[ r = \frac{2T \cos \theta}{h \rho g} = \frac{2 \times 7.5 \times 10^{-2} \times 1}{0.075 \times 1000 \times 10} \]
\[ r = \frac{0.15}{750} = \frac{15 \times 10^{-2}}{7.5 \times 10^{2}} = 2 \times 10^{-4}\text{ m} \]
Diameter \(d = 2r = 2 \times 2 \times 10^{-4} = 4 \times 10^{-4}\text{ m}\).
Convert to mm: \(d = 4 \times 10^{-4} \times 10^{3} = 0.4\text{ mm}\).
Step 4: Final Answer:
The diameter of the capillary tube is \(0.4\text{ mm}\).