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Question #42892

Difficulty: 8/10 | 🟢 Free

The Question

A thin, fixed, non-conducting disk of radius $R \text{ m}$ has a uniform surface charge density $\sigma \text{ C/m}^2$ . A point charge $q \text{ C}$ is placed at a point $P$ which is at a height $z \text{ m}$ above the center of the disk and at a very small radial distance $r \text{ m}$ ( $r \ll R, z$ ) from the central axis of symmetry. Which of the following expressions correctly represents the approximate radial outward force $F_r$ acting on the point charge $q$ ? (Assume vacuum with permittivity $\epsilon_0 \text{ F/m}$ )

Options

A.
$\frac{q \sigma R^2 r}{4\epsilon_0 (R^2+z^2)^{3/2}} \text{ N}$
B.
0 \text{ N}
C.
$\frac{q \sigma z r}{2\epsilon_0 (R^2+z^2)^{3/2}} \text{ N}$
D.
$\frac{q \sigma r}{4\epsilon_0 \sqrt{R^2+z^2}} \text{ N}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Coulomb's Law Vector Form|Symmetry and Virtual Work|Flux and Surface Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from a 1D axial field to a 3D spatial field using the divergence theorem ( $\nabla \cdot \mathbf{E} = 0$ ) in a charge-free region.

The primary trap is assuming that near the center of the disk, the field behaves like an infinite sheet with a uniform field, leading to the incorrect conclusion that the radial force is zero (Option B).

Students who attempt to use Coulomb's law vector summation directly will face an unsolvable elliptic integral, punishing those who rely on brute-force integration over elegant symmetry and local field equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Coulomb's Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free