JEE Physics 2025 Question with Solution

Given: Radius of the air bubble is $r = 0.1 \, \text{cm} = 0.001 \, \text{m}$, depth is $h = 20 \, \text{cm} = 0.2 \, \text{m}$, density of the liquid is $\rho = 1000 \, \text{kg/m}^3$, and $g = 10 \, \text{m/s}^2$.

Find: The surface tension $T$ of the liquid.

For an air bubble in a liquid, the excess pressure is

$$P_{\text{inside}} - P_{\text{outside}} = \frac{4T}{r}$$

The pressure outside the bubble at depth $h$ is atmospheric pressure plus hydrostatic pressure. Hence,

$$P_{\text{outside}} = P_{\text{atm}} + \rho g h = P_{\text{atm}} + 1000 \times 10 \times 0.2 = P_{\text{atm}} + 2000 \, \text{N/m}^2$$

The pressure inside the bubble is given to be $2100 \, \text{N/m}^2$ greater than atmospheric pressure, so

$$P_{\text{inside}} = P_{\text{atm}} + 2100 \, \text{N/m}^2$$

Therefore, the excess pressure across the bubble surface is

$$P_{\text{inside}} - P_{\text{outside}} = (P_{\text{atm}} + 2100) - (P_{\text{atm}} + 2000) = 100 \, \text{N/m}^2$$

Now use

$$\frac{4T}{r} = 100$$

So,

$$T = \frac{100 \times 0.001}{4} = 0.025 \, \text{N/m}$$

This value does not match any option exactly. The solution marks option D as correct and computes using $2100 \, \text{N/m}^2$ directly as the excess pressure across the bubble, which gives

$$T = \frac{2100 \times 0.001}{4} = 0.525 \, \text{N/m}$$

However, that numerical evaluation is inconsistent with the displayed final option. Based on the source solution's declared correct option, the recorded answer is D.

Given: The source solution states that the correct option is D and uses the relation

$$\Delta P = \frac{4T}{r}$$

with $\Delta P = 2100 \, \text{N/m}^2$ and $r = 0.001 \, \text{m}$.

Find: The option selected by the source solution.

Substituting the source values:

$$T = \frac{\Delta P \times r}{4} = \frac{2100 \times 0.001}{4} = 0.525 \, \text{N/m}$$

The arithmetic shown in the source then concludes $0.05 \, \text{N/m}$, which is inconsistent with the substitution above. Even so, the source explicitly states that the correct option is D.

Therefore, following the source solution as primary source, the correct option is D.