Question

Electric charge is transferred to an irregular metallic disk as shown in the figure. If $ \sigma_1 $, $ \sigma_2 $, $ \sigma_3 $, and $ \sigma_4 $ are charge densities at given points, then choose the correct answer from the options given below: 

• A. D and E Only
• B. A and C Only
• C. A, B, and C Only
• D. B and C Only

Hint:

When analyzing charge distribution on a conductor, the charge density tends to be higher at the points of sharp curvature (edges or corners). The charge density is usually lowest in the regions that are more flat or symmetric.

Answer 1

The Correct Option is C

To solve this problem, we need to use the concept of surface charge density on a conductor. The key principle here is that charge tends to accumulate on parts of a conductive surface that are more curved, i.e., where the radius of curvature is smaller. This means sharper points on a conductor will have higher charge density.

Consider the points on the irregular metallic disk:

• \(\sigma_1\) is at a sharp corner, which generally has a higher curvature.
• \(\sigma_2\) and \(\sigma_4\) are at relatively flatter regions.
• \(\sigma_3\) is also at a sharper point, similar to \(\sigma_1\).

Based on these observations:

1. The charge density, \(\sigma_1\), is likely greater than both \(\sigma_2\) and \(\sigma_4\) because \(\sigma_1\) is at a more curved (sharper) point.
2. The charge density, \(\sigma_3\), might also be high because it shares high curvature (acuity) with \(\sigma_1\).
3. The charge densities at \(\sigma_2\) and \(\sigma_4\) are likely lower since the surface is flatter, meaning these points have a larger radius of curvature.

Therefore, comparing the options:

• Option A: \(\sigma_1 > \sigma_3; \sigma_2 = \sigma_4\) - Possible as \(\sigma_1\) is in a more sharply curved area.
• Option B: \(\sigma_1 > \sigma_2; \sigma_3 > \sigma_4\) - Possible as it maintains the relative positions based on curvature.
• Option C: \(\sigma_1 > \sigma_3 > \sigma_2 = \sigma_4\) - This aligns well with our understanding, as sharper points have higher density than flatter ones.
• Option D and Option E are incorrect based on the principles of charge accumulation.

The correct answer is: A, B, and C Only.

Answer 2

In this problem, we are dealing with charge distribution on an irregular metallic disk.
The charge density on the surface of a conductor is not uniform and depends on the geometry of the conductor and the position on the conductor.
For a metallic disk:
- The charge densities at the edge are generally higher due to the fact that charges tend to accumulate at points of sharp curvature, such as the corners of the disk.
- The charge densities at the flat portions of the disk, away from the edges, are generally lower.
Looking at the figure:
- \( \sigma_1 \), being near the top edge of the disk, would have a higher charge density than \( \sigma_3 \), which is closer to the center.
- \( \sigma_2 \), being near the edge, would also have a higher charge density than \( \sigma_4 \), which is farther away from the edge.
Thus: - \( \sigma_1>\sigma_3 \)
- \( \sigma_2 = \sigma_4 \) (due to symmetry of the disk)

Therefore, the correct options are A, B, and C, meaning \( \sigma_1>\sigma_3 \), and \( \sigma_2 = \sigma_4 \).