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Question #43104

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The Question

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$ ( $d \ll R$ ). It is being charged by a constant current $I_0$ starting from an uncharged state at $t = 0$ . Assume the electric field $\vec{E}$ is uniform between the plates and zero outside. Which of the following expressions correctly gives the magnitude of the Poynting vector $S$ at a radial distance $r$ ( $r < R$ ) from the central axis between the plates, and the total electromagnetic power $P_{in}$ flowing into the cylindrical volume of radius $r$ and height $d$ between the plates at time $t$ ?

Options

A.
$S = \frac{I_0^2 t r}{2 \pi^2 \epsilon_0 R^4}, \quad P_{in} = \frac{I_0^2 t d r^2}{\pi \epsilon_0 R^4}$
B.
$S = \frac{I_0^2 t}{2 \pi^2 \epsilon_0 R^2 r}, \quad P_{in} = \frac{I_0^2 t d}{\pi \epsilon_0 R^2}$
C.
$S = \frac{I_0^2 t}{2 \pi^2 \epsilon_0 R^3}, \quad P_{in} = \frac{I_0^2 t d r}{\pi \epsilon_0 R^3}$
D.
$S = \frac{I_0^2 t r^2}{2 \pi^2 \epsilon_0 R^5}, \quad P_{in} = \frac{I_0^2 t d r^3}{\pi \epsilon_0 R^5}$

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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

Poynting Vector and Intensity|Ampere’s Circuital Law Symmetry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from lumped-circuit macroscopic models to rigorous field-theoretic electrodynamics.

A common misconception is that energy flows directly through the wires into the capacitor plates.

If students apply this flawed intuition, they incorrectly use the Biot-Savart law for a long straight wire ( $B \propto 1/r$ ) instead of integrating the time-varying displacement current density ( $B \propto r$ ) via the Ampere-Maxwell law.

This leads to the erroneous conclusion that the power flux is constant across all radii (Option B).

The correct application of the Poynting vector demonstrates that electromagnetic energy actually enters the capacitor radially from the surrounding space, with the power scaling as $r^2$ within the dielectric region.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Poynting Vector and Intensity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free