JEE Physics 2025 Question with Solution

Given: An irregular metallic disk has surface charge densities $\sigma_1, \sigma_2, \sigma_3, \sigma_4$ at the marked points.

Find: Which listed statements about the relative magnitudes of these charge densities are correct.

For a conductor in electrostatic equilibrium, surface charge density is higher at points with smaller radius of curvature, that is, at sharper portions of the conductor. Relatively flatter portions have lower charge density.

From the figure:

• $\sigma_1$ is at a sharp tip, so it is high.
• $\sigma_3$ is also at a sharp point, but less sharp than the top tip, so it is lower than $\sigma_1$.
• $\sigma_2$ and $\sigma_4$ are at corresponding side points on smoother curved portions, so they are equal and lower.

Thus the relation that best matches the figure is

$$\sigma_1 > \sigma_3 > \sigma_2 = \sigma_4$$

Now check the statements shown in the figure:

• A: $\sigma_1 > \sigma_3;\; \sigma_2 = \sigma_4$ — true
• B: $\sigma_1 > \sigma_2;\; \sigma_3 > \sigma_4$ — true
• C: $\sigma_1 > \sigma_3 > \sigma_2 = \sigma_4$ — true

Therefore, the correct option is C, that is, A, B, and C Only.

Given: Charge is placed on an irregular metallic disk.

Find: The correct set of true comparisons among $\sigma_1, \sigma_2, \sigma_3, \sigma_4$.

The key physical idea is that on a conductor, charges crowd more strongly near sharp projections. Hence sharper boundary points have greater surface charge density than smoother side regions.

Using that idea:

1. The top point $\sigma_1$ lies at the sharpest point, so it is the greatest.
2. The bottom point $\sigma_3$ is also pointed, so it is greater than side values.
3. The side points $\sigma_2$ and $\sigma_4$ lie on similar smoother curved portions, so they are equal.

Hence

$$\sigma_1 > \sigma_3 > \sigma_2 = \sigma_4$$

This immediately implies statement A is correct, statement B is correct, and statement C is also correct.

So the final choice is A, B, and C Only, which corresponds to option C.