Before You Begin

Question #43171

Difficulty: 8/10 | 🟢 Free

The Question

A spherical region of radius $R$ in vacuum contains a time-varying, non-uniform volumetric charge density given by $\rho(r,t) = \rho_0 \frac{R}{r} \cos(\omega t)$ for $r \le R$ , where $r$ is the radial distance from the center, $\rho_0$ is a positive constant, and $\omega$ is the angular frequency. Using the standard retarded scalar potential formulation (Lorenz gauge), the electric potential at the center of the sphere can be expressed as $V(0,t) = V_0 \cos(\omega t - \delta)$ . What are the exact expressions for the amplitude $V_0$ and the phase retardation $\delta$ ?

Options

A.
$V_0 = \frac{2 \rho_0 R c}{\epsilon_0 \omega} \sin\left(\frac{\omega R}{2c}\right)$ and $\delta = \frac{\omega R}{2c}$
B.
$V_0 = \frac{\rho_0 R^2}{\epsilon_0}$ and $\delta = \frac{2\omega R}{3c}$
C.
$V_0 = \frac{\rho_0 R^2}{\epsilon_0}$ and $\delta = \frac{\omega R}{c}$
D.
$V_0 = \frac{\rho_0 R^2}{\epsilon_0}$ and $\delta = 0$

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

Phase Relationship|Gauss Law Applicability|Non-conducting Charged Sphere

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the Liénard–Wiechert retarded potential formulation to an extended, non-uniform source.

A common trap is to rely on static Gauss's Law intuition, which incorrectly predicts zero phase lag (Option D).

Another sophisticated trap is to assume the phase lag is determined by the 'center of charge' or average distance of the charge distribution.

For $\rho \propto 1/r$ , the average distance is $\langle r \rangle = 2R/3$ , leading students to incorrectly guess $\delta = 2\omega R/3c$ (Option B).

By requiring the exact integration of the retarded potential, the problem demonstrates that the $1/r$ dependence of the potential exactly cancels the $r$ dependence of the charge density, resulting in a uniform spatial weighting and an exact phase retardation of $R/2c$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Phase Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free