Question #43021

This question tests the synthesis of the Biot-Savart Law with Kepler's Second Law (Conservation of Angular Momentum) in a non-circular orbit.

A common trap is for students to treat the elliptical orbit as an equivalent circular current loop $I = q/T$ and incorrectly guess an effective radius, such as the semi-major axis $a$ or semi-minor axis $b = a\sqrt{1-e^2}$.

This leads to incorrect options like $\frac{\mu_0 q}{2 T a}$ or $\frac{\mu_0 q}{2 T a \sqrt{1-e^2}}$.

The lethal collision occurs when students fail to realize that the magnetic field varies as $1/r^3$ while the differential time element $dt$ varies as $r^2 d\theta$.

By applying Kepler's Second Law to substitute $dt$, the time integral elegantly transforms into an angle integral of $1/r$, revealing that the effective radius for the time-averaged magnetic field is exactly the semi-latus rectum $a(1-e^2)$.