Chapter 27 Problem 74

Problem

Current loop and varying B

The net force on a current loop whose face is perpendicular to a uniform magnetic field is zero, since contributions to the net force from opposite sides of the loop cancel. However, if the field varies in magnitude from one side of the loop to the other, then there can be a net force on the loop. Consider a square loop with sides whose length is $a$, located with one side at $x=b$ in the $xy$ plane. A magnetic field is directed along $z$, with a magnitude that varies with $x$ according to

$B=B_{0}\left(1-{\frac {x}{b}}\right)$

If the current in the loop circulates counterclockwise (that is, the magnetic dipole moment of the loop is along the $z$ axis), find an expression for the net force on the loop.

solution

The forces on each of the two horizontal sides of the loop have the same magnitude, but opposite directions, so these forces sum to zero. The left side of the loop is located at $x=b$, where the magnetic field is zero, and therefore the force is zero. The net force is the force acting on the right side of the loop. By the right hand rule, with the current directed toward the top of the page and the magnetic field into the page, the force will point in the negative $x$ direction with magnitude

${\vec {F}}=IlB(-{\hat {i}})=IaB_{0}\left(1-{\frac {b+a}{b}}\right){\hat {i}}=-{\frac {Ia^{2}B_{0}}{b}}{\hat {i}}$