Before You Begin

Question #43309

Difficulty: 8/10 | 🟢 Free

The Question

A small circular coil of radius $a$ and $N$ turns is placed parallel to the $xy$ -plane at a height $z = d$ above a perfectly conducting infinite plane located at $z = 0$ . An alternating current $I(t) = I_0 \cos(\omega t)$ flows through the coil. The frequency $\omega$ is such that $d = \lambda/3$ , where $\lambda$ is the wavelength of the emitted electromagnetic radiation. A small secondary receiver is placed at a distance $R$ ( $R \gg d, \lambda$ ) from the origin, such that its position vector makes an angle $\theta = \pi/3$ with the positive $z$ -axis. Assuming the coil acts as an ideal magnetic dipole and the space above the plane is vacuum, what is the time-averaged intensity of the electromagnetic radiation at the receiver's location?

Options

A.
$\frac{9 \mu_0 N^2 a^4 I_0^2 \omega^4}{128 c^3 R^2}$
B.
$\frac{3 \mu_0 N^2 a^4 I_0^2 \omega^4}{64 c^3 R^2}$
C.
$\frac{3 \mu_0 N^2 a^4 I_0^2 \omega^4}{128 c^3 R^2}$
D.
$\frac{9 \mu_0 N^2 a^4 I_0^2 \omega^4}{64 c^3 R^2}$

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

Mutual Inductance Reciprocity|Poynting Vector and Intensity|Boundary Condition Reflection|Maxwell's Equations

Why This Question Is Tricky

Common Mistakes

This question subverts the 'Quasi-static Bias' and tests the correct application of boundary conditions for magnetic dipoles.

Students often incorrectly assume that the image of a magnetic dipole perpendicular to a conducting plane points in the same direction (like an electric dipole), ignoring that a parallel current loop's image must have reversed current to cancel the normal magnetic field.

If a student uses a non-inverted image, they will calculate the interference term as $\cos^2(kd\cos\theta)$ instead of $\sin^2(kd\cos\theta)$ , leading directly to Option C.

Furthermore, students who ignore retardation entirely or add intensities incoherently will land on Option B.

The problem forces the rigorous synthesis of image theory, retarded potentials, and Poynting vector intensity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Mutual Inductance Reciprocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free