Question #43401

This problem tests the synthesis of electromagnetism and classical orbital mechanics.

A common trap is to treat the elliptical orbit as an equivalent circular current loop of radius $R = \sqrt{ab}$ (matching the area) or $R = a$, which incorrectly assumes a constant angular velocity and uniform distance.

Students often fail to recognize that the instantaneous magnetic field depends on $1/r^2$ while the angular velocity also depends on $1/r^2$, making the time-average heavily weighted towards the periapsis.

By applying Kepler's Second Law ($dt \propto r^2 d\theta$), the $r^3$ term in the Biot-Savart denominator partially cancels, transforming the time integral into a purely geometric angular integral of $1/r$.

This elegantly avoids the need for complex time-dependent parameterizations and exposes the fallacy of the circular loop approximation.