Question

A cone made of conducting material is given a charge $ Q $. $ \sigma_1, \sigma_2, \sigma_3 $ and $ \sigma_4 $ are charge densities at four points $ P, Q, R $ and $ S $. $ P $ is at the vertex of the cone and $ Q, R, S $ are at the periphery of the base. Choose the correct option.

• A. \( \sigma_1 > \sigma_2 > \sigma_3 > \sigma_4 \)
• B. \( \sigma_1 > \sigma_2 = \sigma_3 = \sigma_4 \)
• C. \( \sigma_1 < \sigma_2 = \sigma_3 < \sigma_4 \)
• D. \( \sigma_1 = \sigma_2 > \sigma_3 > \sigma_4 \)

Hint:

In conducting materials, charge density is inversely related to the surface area. The sharpest regions (like the vertex of a cone) have the highest charge density.

Answer 1

The Correct Option is B

In conductive materials, charge density is maximized at points of greatest curvature, such as the apex of a cone, and diminishes with increasing surface area, as observed towards the base.
The charge density at the apex, denoted by \( \sigma_1 \), is the highest.
Charge densities at points \( Q, R, S \), which are symmetrically positioned at the base's periphery, are equivalent due to their larger surface area relative to the apex.
Consequently, the charge density relationship is established as:\[\sigma_1 > \sigma_2 = \sigma_3 = \sigma_4\]
Final Answer: The correct option is (2), \( \sigma_1 > \sigma_2 = \sigma_3 = \sigma_4 \).