Before You Begin

Question #43887

Difficulty: 8/10 | 🟢 Free

The Question

A thin spherical shell of equilibrium radius $R_0$ carries a uniformly distributed total charge $Q$ . The shell undergoes small radial pulsations such that its radius at time $t$ is given by $R(t) = R_0 + a \sin(\omega t)$ , where $a \ll R_0$ . The pulsations are perfectly spherically symmetric. Let $\langle P \rangle$ be the time-averaged total electromagnetic power radiated by the shell across a large spherical surface of radius $r \gg R_0$ . Which of the following expressions gives $\langle P \rangle$ ?

Options

A.
$0$
B.
$\frac{\mu_0 Q^2 a^2 \omega^4}{12 \pi c}$
C.
$\frac{\mu_0 Q^2 a^2 \omega^4}{8 \pi c}$
D.
$\frac{\mu_0 Q^2 a^2 \omega^4}{6 \pi c}$

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

Wave Propagation Direction|Electric Field Symmetry|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of radiation formulas (such as Larmor's formula) to highly symmetric extended sources.

A common trap is assuming that because the charges on the shell are accelerating radially, they must radiate electromagnetic waves with a $1/r$ spatial dependence.

Students often incorrectly apply point-charge or plane-wave assumptions without analyzing the global boundary conditions imposed by the source geometry.

Rigorous application of Maxwell's equations reveals that for a perfectly spherically symmetric pulsating charge, Gauss's Law forces the external electric field to be static, and Ampere's Law (combined with symmetry) forces the magnetic field to be identically zero.

Consequently, the Poynting vector is zero, and no radiation occurs.

This is a fundamental consequence of the fact that electromagnetic fields do not support monopole radiation.

Quick Facts About This Question

Question Information

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