Before You Begin

Question #43257

Difficulty: 8/10 | 🟢 Free

The Question

An electron of mass $m$ (in kg) and charge $-e$ (in C) orbits a massive nucleus in an elliptical orbit. The orbit has a semi-major axis $a$ (in m), eccentricity $\epsilon$ (dimensionless), and time period $T$ (in s). The nucleus is located at one of the foci of the ellipse. What is the magnitude of the instantaneous magnetic field $B$ (in T) produced by the electron at the nucleus when the electron is at its apocenter (farthest point from the nucleus)? Assume non-relativistic motion and vacuum permeability $\mu_0$ (in $\text{T}\cdot\text{m/A}$ ).

Options

A.
$\frac{\mu_0 e}{2 a T} \frac{\sqrt{1-\epsilon}}{(1+\epsilon)^{5/2}}$
B.
$\frac{\mu_0 e}{2 a T} \frac{1}{1+\epsilon}$
C.
$\frac{\mu_0 e}{2 a T} \frac{1}{(1+\epsilon)^2}$
D.
$\frac{\mu_0 e}{2 a T} \sqrt{\frac{1-\epsilon}{1+\epsilon}}$

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

Magnetic Dipole Moment of Revolving Electron|Kepler’s Second Law (Areal Velocity)|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of the average current formula $I = q/T$ to calculate instantaneous magnetic fields in non-circular orbits.

Students often fall into the trap of assuming the electron acts as a steady circular current loop of radius $r = a(1+\epsilon)$ , leading to the incorrect Biot-Savart application $B = \mu_0 I / (2r)$ (Option B).

Another common error is assuming a constant average orbital speed $v = 2\pi a/T$ (Option C).

The correct physical framework requires recognizing that while the linear and angular velocities fluctuate, Kepler's Second Law dictates that the areal velocity remains constant.

This intrinsically links the instantaneous magnetic field to the conserved orbital angular momentum, demanding a derivation from fundamental point-charge electrodynamics rather than macroscopic steady-current approximations.

Quick Facts About This Question

Question Information

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