Question

One end of a capillary tube is dipped in water, the rise of water column is '\(h\)'. The upward force of 98 dyne due to surface tension is balanced by the force due to the weight of the water column. The inner circumference of the capillary is ( surface tension of water \(= 7 \times 10^{-2}\text{Nm}^{-1}\) )

• A. \(1.4\text{ cm}\)
• B. \(0.7\text{ cm}\)
• C. \(0.14\text{ cm}\)
• D. \(0.07\text{ cm}\)

Hint:

Always convert dyne to Newton: \(1\text{ dyne} = 10^{-5}\text{ N}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The upward force in a capillary tube is provided by surface tension acting along the line of contact.
Assuming the contact angle for water and glass is zero (\(\theta = 0\)), the surface tension force acts vertically upward along the entire inner circumference.
Step 2: Key Formula or Approach:
Surface tension force \(F = T \times L\), where \(L\) is the total length along which the tension acts.
For a circular capillary, \(L\) is the inner circumference, \(2\pi r\).
Step 3: Detailed Explanation:
Given: \(F = 98\text{ dyne}\). Convert it to SI units: \[ 1\text{ dyne} = 10^{-5}\text{ N} \implies F = 98 \times 10^{-5}\text{ N} \] Surface tension \(T = 7 \times 10^{-2}\text{ N/m}\).
Substitute these into the formula \(F = T \cdot (\text{Circumference})\): \[ 98 \times 10^{-5} = (7 \times 10^{-2}) \times \text{Circumference} \] Solve for Circumference: \[ \text{Circumference} = \frac{98 \times 10^{-5}}{7 \times 10^{-2}} \] \[ \text{Circumference} = 14 \times 10^{-3}\text{ m} \] Convert meters to centimeters: \[ \text{Circumference} = 14 \times 10^{-3} \times 100\text{ cm} = 1.4\text{ cm} \] Step 4: Final Answer:
The inner circumference of the capillary tube is \(1.4\text{ cm}\).