Torque on a Current Loop — Question 2

Question 2

Opposite side forces form a couple with zero net force and nonzero torque

A rectangular loop centered at the origin lies in the xy plane, with side a along x and side b along y. Current I is counterclockwise viewed from +z. The uniform field is $B\hat x$.

1. Find the forces on all four sides and their vector sum.

2. Find the torque about the center and show that changing the torque origin does not change it.

Question 2 - Solution

Opposite side forces form a couple with zero net force and nonzero torque

(a) Method and calculation

$$\vec F_{x=+a/2}=-IbB\hat z,\quad\vec F_{x=-a/2}=+IbB\hat z;\quad\vec F_{\mathrm{horizontal}}=0.$$$$\sum\vec F=0.$$

(b) Interpretation and further calculation

$$\vec\tau=\frac a2\hat x\times(-IbB\hat z)-\frac a2\hat x\times(IbB\hat z)=IabB\hat y.$$$$\vec\tau\prime=\vec\tau-\vec r_0\times\sum\vec F=\vec\tau.$$

Physical check

A force couple produces rotation without accelerating the center of mass in a uniform field.