Before You Begin

Question #42906

Difficulty: 8/10 | 🟢 Free

The Question

Two concentric thin conducting spherical shells have radii $R$ and $2R$ . The inner shell is grounded, and the outer shell is given a total charge $+Q$ . A small electric dipole of dipole moment magnitude $p$ is placed in the region between the shells at a distance $r = \frac{3}{2}R$ from their common center. The dipole is held fixed such that its dipole moment vector $\vec{p}$ makes an angle of $60^\circ$ with the outward radial vector at its location. What is the magnitude of the net electric force experienced by the dipole? (Assume the presence of the dipole does not alter the charge distribution on the shells)

Options

A.
$\frac{\sqrt{7} pQ}{54\pi\epsilon_0 R^3}$
B.
$\frac{pQ}{27\pi\epsilon_0 R^3}$
C.
$\frac{\sqrt{7} pQ}{27\pi\epsilon_0 R^3}$
D.
$\frac{\sqrt{3} pQ}{54\pi\epsilon_0 R^3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Dipole in Non-Uniform Field|Potential of Concentric Shells|Potential Gradient Direction

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the dipole force equation $\vec{F} = (\vec{p} \cdot \nabla)\vec{E}$ in a non-uniform field.

The primary trap is assuming the force is simply the gradient of the potential energy $F = -\nabla(-\vec{p} \cdot \vec{E})$ while treating the angle between $\vec{p}$ and $\vec{E}$ as constant, which incorrectly yields only a radial force component.

Students often fail to recognize that a radial field exerts a tangential force on a dipole with a tangential component, because the direction of the radial field changes as one moves tangentially ( $\frac{\partial \hat{r}}{\partial \theta} = \hat{\theta}$ ).

Additionally, it tests the concentric shell potential concept where grounding the inner shell induces a charge $-Q/2$ , not $-Q$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dipole in Non-Uniform Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free