Before You Begin

Question #43014

Difficulty: 8/10 | 🟢 Free

The Question

An infinitely long solenoid of radius $a$ has $n$ turns per unit length and carries a steady current $I$ . The interior of the solenoid ( $r < a$ ) is filled with a non-conducting magnetic material whose relative permeability varies radially as $\mu_r(r) = 1 + \alpha r^2$ , where $r$ is the distance from the solenoid axis and $\alpha$ is a positive constant. A particle of mass $m$ and charge $q$ ( $q > 0$ ) is injected into the medium such that it moves in a steady circular orbit of radius $R$ ( $R < a$ ) centered on the solenoid axis, in a plane perpendicular to the axis. Neglecting any energy losses, what is the kinetic energy of the particle required to maintain this orbit?

Options

A.
$\frac{q^2 \mu_0^2 n^2 I^2 R^2}{2m} \left(1 + \alpha R^2\right)^2$
B.
$\frac{q^2 \mu_0^2 n^2 I^2 R^2}{2m}$
C.
$\frac{q^2 \mu_0^2 n^2 I^2 R^2}{2m} \left(1 + \frac{\alpha R^2}{2}\right)^2$
D.
$\frac{q^2 \mu_0^2 n^2 I^2 R^2}{2m} \left(1 + \alpha R^2\right)$

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

Cyclotron Frequency Invariance|Gauss’s Law for Magnetism|Magnetic Permeability and Susceptibility

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from macroscopic vacuum formulas to localized fields in non-uniform media.

The primary trap is assuming the magnetic field inside the solenoid is either uniform ( $B = \mu_0 n I$ ) or governed by the average flux enclosed by the orbit.

By introducing a radially varying permeability $\mu_r(r)$ , students must recognize that the auxiliary field $\mathbf{H}$ remains uniform due to the free current distribution, while $\mathbf{B}$ varies locally.

The cyclotron frequency and required centripetal force depend strictly on the local magnetic field $B(R)$ at the orbit's radius, not the vacuum field or the spatial average.

Applying the vacuum cyclotron formula leads to a fatal error.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Cyclotron Frequency Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free