Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by 75 x 10⁻⁴ N force due to the weight of the liquid. If the surface tension of water is 6 x 10⁻² N / m, the inner circumference of the capillary must be

The correct option is 12.5 x 10⁻² m.

Understanding the Concept:
When a capillary tube is dipped in a liquid like water, the liquid rises due to surface tension. The upward force supporting the liquid column is provided by the vertical component of the surface tension force acting along the line of contact. For a liquid that wets the capillary tube completely (such as water in glass, where the contact angle $\theta \approx 0$), the surface tension force acts vertically upwards along the entire inner circumference of the tube.

The total upward force $F$ due to surface tension is given by:
$F=T\times C$
where:
• $T$ is the surface tension of the liquid.
• $C$ is the inner circumference of the capillary tube.

At equilibrium, this upward force balances the downward force due to the weight of the liquid column $W$:
$T\times C=W$

Given Data:
• Force due to the weight of the liquid, $W=75\times {10}^{-4}\text{N}$
• Surface tension of water, $T=6\times {10}^{-2}\text{N/m}$

Step-by-Step Calculation:
We can rearrange the equilibrium formula to solve for the inner circumference $C$:
$C=\frac{W}{T}$

Substitute the given values into the equation:
$C=\frac{75\times {10}^{-4}}{6\times {10}^{-2}}$

Simplify the fraction:
$C=\left(\frac{75}{6}\right)\times {10}^{-4-(-2)}$

$C=12.5\times {10}^{-2}\text{m}$

Thus, the inner circumference of the capillary tube must be $12.5\times {10}^{-2}\text{m}$.