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Question #43161

Difficulty: 8/10 | 🟢 Free

The Question

An ideal oscillating electric dipole with dipole moment $\vec{p}(t) = p_0 \sin(\omega t) \hat{k}$ is placed at the origin. The surrounding region also contains a uniform time-varying electric field $\vec{E}_c(t) = E_0 \sin(\omega t) \hat{k}$ produced by external sources. Consider a circular Amperian loop of radius $R$ in the $xy$ -plane, centered at the origin. To apply the Ampere-Maxwell law without encountering the conduction current singularity of the point dipole at the origin, a student chooses the open surface $S$ bounded by the loop to be the upper hemisphere ( $x^2 + y^2 + z^2 = R^2$ , $z > 0$ ). What is the total displacement current $I_d$ passing through the surface $S$ in the outward radial direction $\hat{r}$ ?

Options

A.
$\left( \pi \epsilon_0 R^2 E_0 + \frac{p_0}{2R} \right) \omega \cos(\omega t)$
B.
$\left( \pi \epsilon_0 R^2 E_0 - \frac{p_0}{4R} \right) \omega \cos(\omega t)$
C.
$\left( \pi \epsilon_0 R^2 E_0 + \frac{p_0}{R} \right) \omega \cos(\omega t)$
D.
$\pi \epsilon_0 R^2 E_0 \omega \cos(\omega t)$

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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

Displacement Current Source|Electric Field Symmetry|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Ampere-Maxwell law and electric flux integration over non-planar surfaces.

A common trap (Option B) is the 'infinite plane approximation', where students incorrectly calculate the flux by multiplying the area of the flat circular disk by the dipole's electric field evaluated at the equatorial edge ( $\vec{E} = -p_0/(4\pi\epsilon_0 R^3)\hat{k}$ ).

This ignores the non-uniformity of the dipole field and the singularity at the origin.

Another trap (Option D) is assuming the dipole's flux through the hemisphere is zero due to symmetry.

By forcing the use of a hemispherical surface, the problem evaluates the student's ability to integrate the radial component of a dipole's field and correctly apply the superposition principle for displacement current.

Quick Facts About This Question

Question Information

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