A long wire is bent to form a rectangular section and is placed in a region with a uniform magnetic field, as shown in figure. The magnetic field is directed downward and has a magnitude of 2.5 T. The wire carries a 25-A current. Find the net torque on the wire about the z-axis

Explanation

We will find the torque by finding the magnetic force on each segment of wire and then finding the torque due to each segment. The torques will sum to the net torque. There are five segments of wire: two lying along the z-axis, two vertical segments, and the 1.0m long horizontal segment that crosses the y-axis. the two segments lying along the z-axis have a net force acting on them, but no torque, since the moment arm about them is zero. The two vertical segments have no force acting on them, because the segments are parallel to the magnetic field. The remaining segments is the horizontal segment at the center. There is a net force, as well as a torque, acting on this segment. Figure provides a view form the z-axis. We see that the force is directed to the right, due to the right-hand rule. The torque is along the negative z-axis and will tend to rotate the segment clockwise.

We find the magnitude of the torque on the horizontal segment by first finding the magnitude of the force on that segment. The magnetic force on a segment of current -carrying wire is
`F = ilB sin phi`
In this case, the current is perpendicular to the magnetic field, so `phi = 90^(@) and sin phi = 1`. The distance between the z-axis and the force is 0.25m, and the force is perpendicular to the moment arm, so the torque is
`tau = rF`
`= ilB`
`= (0.25 m) (25A) (1.0m) (2.5T)`
= 15.6 N m
The net force on the wire is 15.6 Nm, directed along the negative z-axis.