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Question #43428

Difficulty: 8/10 | 🟢 Free

The Question

Two identical circular coils, each of radius $R$ and having $N$ turns, are placed coaxially at a distance $R$ apart in a Helmholtz configuration. A time-varying current $I(t) = k t$ (where $k$ is a positive constant in $\text{A s}^{-1}$ ) flows through both coils in the same direction. An electron (mass $m$ , charge $-e$ ) is initially at rest at a small distance $r_0$ ( $r_0 \ll R$ ) from the common axis, in the mid-plane between the coils. Assuming the radial displacement remains negligible ( $r \approx r_0$ ), what is the radial acceleration $a_r$ of the electron at time $t$ ? (Assume non-relativistic mechanics and ignore the self-inductance of the electron's orbit).

Options

A.
$-\frac{16 e^2 \mu_0^2 N^2 k^2 t^2 r_0}{125 m^2 R^2}$
B.
$\frac{16 e^2 \mu_0^2 N^2 k^2 t^2 r_0}{125 m^2 R^2}$
C.
$-\frac{e^2 \mu_0^2 N^2 k^2 t^2 r_0}{16 m^2 R^2}$
D.
$-\frac{32 e^2 \mu_0^2 N^2 k^2 t^2 r_0}{125 m^2 R^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Magnetic Moment of Orbiting Electron|Helmholtz Coils Configuration|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple Faraday's Law of Induction with the Lorentz force and circular kinematics in a non-steady state.

The primary trap lies in the radial equation of motion.

Students often correctly calculate the induced azimuthal velocity (Larmor frequency) but incorrectly assume the orbit remains stable ( $a_r = 0$ ) or forget to subtract the centripetal acceleration requirement from the inward magnetic Lorentz force.

By doing so, they either conclude the radial acceleration is zero or incorrectly identify it as purely centrifugal (outward) or purely magnetic (missing the crucial $1/4$ kinematic factor).

Furthermore, it rigorously tests the exact Biot-Savart derivation for the Helmholtz configuration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Moment of Orbiting Electron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free