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Question #42913

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The Question

A solid sphere of radius $R$ has a non-uniform volume charge density $\rho(r) = \alpha r$ , where $\alpha$ is a positive constant and $r$ is the radial distance from the center. The sphere is cut into two identical hemispheres by a plane passing through its center. The two hemispheres are slowly separated by a very small distance $\delta$ ( $\delta \ll R$ ) by an external agent. Let $\Delta W$ be the work done by the external agent.

A magnetic dipole of magnetic moment $m$ is placed at the origin of a separate coordinate system. The magnetic energy density at a distance $R$ from the dipole on its equatorial plane is $u_B$ .

If the ratio $\frac{\Delta W}{\delta}$ can be expressed as $A \cdot u_B$ , where $A$ is an effective area, find the expression for $A$ . (Assume the separation $\delta$ is small enough that the field configuration outside the gap remains approximately unchanged, and $c$ is the speed of light in vacuum).

Options

A.
$\frac{4 \pi^3 \alpha^2 R^{12} c^2}{3 m^2}$
B.
$\frac{\pi^3 \alpha^2 R^{12} c^2}{3 m^2}$
C.
$\frac{\pi^3 \alpha^2 R^{12} c^2}{m^2}$
D.
$\frac{3 \pi^3 \alpha^2 R^{12} c^2}{2 m^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Continuous Charge Distribution|Energy Density Locality|Effective Moment of Cut Magnets|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the locality of electrostatic energy density and the application of the Work-Energy theorem via the Maxwell Stress Tensor.

The 'Lethal Collision' occurs when students calculate the work done to separate the hemispheres by integrating the energy density only over the gap inside the charge distribution ( $r \le R$ ), assuming that since there is no charge outside, the gap outside doesn't contribute to the energy change.

However, the electric field extends to infinity, and separating the hemispheres creates a gap in the field from $r=0$ to $r=\infty$ .

Failing to account for the energy stored in the gap outside the sphere leads to exactly $1/4$ of the correct force, which corresponds to the trap Option B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Continuous Charge Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free