Question #43269

Students often erroneously assume that magnetic fields, analogous to electrostatic fields, can be represented by a scalar potential gradient ($\vec{B} = -\nabla V_m$), which would imply a path-independent field where $\oint \vec{B} \cdot d\vec{l} = 0$.

This trap exploits that misconception by presenting a superimposed polynomial field with zero divergence but a non-zero curl ($\nabla \times \vec{B} \neq 0$).

If a student assumes the field is conservative or mistakenly calculates the magnetic flux ($\int \vec{B} \cdot d\vec{S} = 0$) instead of the circulation, they will incorrectly conclude the net current is zero.

The problem forces the rigorous application of Ampere's Law, demonstrating that a non-conservative magnetic field necessitates a local current density, thus testing the fundamental distinction between scalar potential applicability and vector field circulation.