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Question #42317

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ carries a uniform volume charge density $\rho$ . A spherical cavity of radius $R/3$ is excavated from the sphere such that the center of the cavity is at a distance $2R/3$ from the center of the solid sphere. Let $V_O$ be the electrostatic potential at the center of the solid sphere and $V_C$ be the electrostatic potential at the center of the cavity. Assuming the potential at infinity to be zero, find the ratio $\frac{V_O}{V_C}$ .

Options

A.
$\frac{13}{10}$
B.
$\frac{26}{23}$
C.
$\frac{5}{4}$
D.
$\frac{10}{7}$

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

Conservative Force Field|Superposition Principle|Vector Calculus|Gradient of Potential

Why This Question Is Tricky

Common Mistakes

This question tests the principle of superposition for electrostatic potentials in non-uniform geometries.

A common trap is for students to incorrectly assume the cavity is an equipotential region (confusing non-conducting with conducting spheres) or to blindly apply the uniform cavity field formula $\vec{E} = \frac{\rho \vec{d}}{3\varepsilon_0}$ to find the potential difference via $\Delta V = -\vec{E} \cdot \Delta \vec{r}$ .

However, this uniform field only exists strictly inside the cavity.

Since the center of the solid sphere lies outside the cavity, the electric field is not uniform along the entire path between the two centers, making the simple line integral invalid and forcing the rigorous use of scalar potential superposition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Conservative Force Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free