Question #42306

This question tests a lethal trap in vector calculus applied to electromagnetism: incorrectly simplifying the surface integral of the magnetic flux by projecting a curved surface onto a single coordinate plane.

In a non-uniform field where the angle between $\vec{B}$ and $d\vec{A}$ varies continuously, students often mistakenly calculate only $\int B_x dA_x$ or $\int B_y dA_y$, which leads directly to the distractors $\ln(2.5)$ or $\ln(1.6)$.

Another common error is ignoring the surface geometry entirely and treating it as a flat radial plane, leading to the $\ln 2$ distractor.

The rigorous approach requires parameterizing the surface to find the exact $d\vec{A}$ vector and evaluating the full dot product.

Alternatively, top-tier students can bypass the trap entirely by using the magnetic vector potential $\vec{A} = -\frac{\mu_0 I}{2\pi} \ln r \hat{k}$ and applying Stokes' Theorem to evaluate the flux as a simple line integral along the boundaries.