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Question #42909

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a non-uniform volume charge density given by $\rho(r) = \rho_0 \left( 1 - \frac{r^2}{R^2} \right)$ , where $r$ is the radial distance from the center of the sphere and $\rho_0$ is a positive constant. A spherical cavity of radius $a = \frac{R}{4}$ is created within the sphere. The center of the cavity is located at a distance $r_0 = \frac{R}{2}$ from the center of the main sphere. What is the magnitude of the electric field at the center of the cavity?

Options

A.
$\frac{21 \rho_0 R}{160 \epsilon_0}$
B.
$\frac{17 \rho_0 R}{120 \epsilon_0}$
C.
$\frac{73 \rho_0 R}{480 \epsilon_0}$
D.
$\frac{11 \rho_0 R}{80 \epsilon_0}$

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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|Superposition Principle|Nuclear Radius Scaling|Integration

Why This Question Is Tricky

Common Mistakes

The core trap in this problem lies in the misapplication of the Superposition Principle for non-uniform continuous charge distributions.

Students frequently memorize that the electric field at the center of a spherical cavity is simply the field produced by the main sphere at that point, implicitly assuming that the 'removed' charge produces zero field at its own center.

While this is true for uniform charge densities (where the removed charge is spherically symmetric about the cavity center), it fails catastrophically for non-uniform densities.

Because the original charge density $\rho(r)$ has a radial gradient, the removed charge possesses a dipole-like asymmetry (a linear density gradient) relative to the cavity's center.

Consequently, the removed charge generates a non-zero electric field at its own center, which must be rigorously calculated and subtracted from the full sphere's field.

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