Physics Solution – Question 9

Solution to Question 9

1. Analyzing the Field Representations

The question asks to identify the incorrect representations of electrostatic field lines. We evaluate each figure based on the properties of conservative electrostatic fields, Gauss’s Law, and boundary conditions for conductors.

2. Analysis of Figure (a): Finite Charged Layers

Observation: The figure shows two finite parallel plates with straight, parallel electric field lines strictly confined between them.

Why it is Incorrect: Electrostatic fields are conservative, meaning the work done in a closed path must be zero: $\oint \vec{E} \cdot d\vec{l} = 0$.

Consider a rectangular loop in the region in between plates. The field strength (represented by density of field lines) above and below are unequal. Hence, the total work done in a closed path is non-zero ($W \neq 0$).

Status: Incorrect

3. Analysis of Figure (b): Point Charge in Shell

Observation: The field lines inside the cavity are curved, and they meet the conducting surface at non-90° angles.

Why it is Incorrect:

Curvature: In a charge-free region (the empty space in the cavity), field lines emanating from a single point charge must be straight radial lines. They cannot curve without external influence.
Orthogonality: The conducting shell is an equipotential surface. Electrostatic field lines must always intersect a conductor at right angles ($90^\circ$).

Status: Incorrect

4. Analysis of Figure (c): Flux Conservation

Observation: We must compare the electric flux entering the inner surface of the shell with the flux exiting the outer surface.

Why it is Incorrect: Consider a Gaussian surface enclosing the entire system. Since the conducting shell is neutral, the total charge enclosed is just the central point charge $+Q$. By Gauss’s Law, the net flux leaving the system must be $\Phi = Q/\epsilon_0$.

This means the number of field lines originating from the center charge must exactly equal the number of field lines emerging from the outer surface of the shell. In diagram (c), a careful count reveals that the number of lines entering the shell is not equal to the number of lines exiting it. This violates the principle of conservation of flux.

Status: Incorrect

5. Analysis of Figure (d): Two Unequal Point Charges

Observation: The figure shows field lines for two positive point charges. The lines repel each other, forming a neutral point (N) between them. The asymmetry of the lines and the position of the neutral point correctly suggest two unequal charges ($q_1 \neq q_2$).

This qualitative representation is physically consistent.

Status: Correct

The incorrect representations are (a), (b), and (c). Only (d) is correct.

Summary of Errors:

(a): Non uniform field between plates, leading to non-zero work in a closed loop.
(b): Shows curved field lines in free space and non-orthogonal intersection with a conductor.
(c): Violates Gauss’s Law/Flux conservation (Flux In $\neq$ Flux Out).