Magnetic Force on a Current-Carrying Wire — Question 7

Question 7

A closed rectangular loop acquires net force in a spatially varying field

A rectangular loop spans 0<x<a and 0<y<b and carries counterclockwise current I viewed from +z. The external field is $\vec B(x)=(B_0+gx)\hat z$. Assume the entire loop lies where this prescribed field model holds.

1. Find the forces on the two vertical sides and show how the horizontal-side forces combine.

2. Find the total force and its direction for g positive; explain why the uniform-field cancellation no longer applies.

Question 7 - Solution

A closed rectangular loop acquires net force in a spatially varying field

(a) Method and calculation

$$\vec F_{\mathrm{right}}=Ib(B_0+ga)\hat x,\quad\vec F_{\mathrm{left}}=-IbB_0\hat x.$$$$\vec F_{\mathrm{bottom}}=-I\int_0^a(B_0+gx)dx\,\hat y=-\vec F_{\mathrm{top}}.$$

(b) Interpretation and further calculation

$$\vec F_{\mathrm{total}}=Iabg\hat x.$$$$g>0\Rightarrow\vec F_{\mathrm{total}}\text{ points toward increasing }B_z.$$

Physical check

Opposite sides sample different magnetic fields, so their equal currents do not imply equal opposing forces.