Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process. (See [link].)

Let us examine the force on each segment of the loop in [link] to find the torques produced about the axis of the vertical shaft. (This will lead to a useful equation for the torque on the loop.) We take the magnetic field to be uniform over the rectangular loop, which has width ww and height ll. First, we note that the forces on the top and bottom segments are vertical and, therefore, parallel to the shaft, producing no torque. Those vertical forces are equal in magnitude and opposite in direction, so that they also produce no net force on the loop. [link] shows views of the loop from above. Torque is defined as τ=rFsinθτ=rFsinθ size 12{τ= ital “rF””sin”θ} {}, where FF size 12{F} {} is the force, rr is the distance from the pivot that the force is applied, and θθ is the angle between rr and FF. As seen in [link](a), right hand rule 1 gives the forces on the sides to be equal in magnitude and opposite in direction, so that the net force is again zero. However, each force produces a clockwise torque. Since r=w/2r=w/2, the torque on each vertical segment is (w/2)Fsinθ(w/2)Fsinθ, and the two add to give a total torque.

Now, each vertical segment has a length ll size 12{l} {} that is perpendicular to BB size 12{B} {}, so that the force on each is F=IlBF=IlB size 12{F= ital “IlB”} {}. Entering FF size 12{F} {} into the expression for torque yields

If we have a multiple loop of NN size 12{N} {} turns, we get NN size 12{N} {} times the torque of one loop. Finally, note that the area of the loop is A=wlA=wl size 12{A= ital “wl”} {}; the expression for the torque becomes

This is the torque on a current-carrying loop in a uniform magnetic field. This equation can be shown to be valid for a loop of any shape. The loop carries a current II size 12{I} {}, has NN size 12{N} {} turns, each of area AA size 12{A} {}, and the perpendicular to the loop makes an angle θθ size 12{θ} {} with the field BB size 12{B} {}. The net force on the loop is zero.

The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at θ=0θ=0 size 12{θ=0} {}. The torque then reverses its direction once the coil rotates past θ=0θ=0 size 12{θ=0} {}. (See [link](d).) This means that, unless we do something, the coil will oscillate back and forth about equilibrium at θ=0θ=0 size 12{θ=0} {}. To get the coil to continue rotating in the same direction, we can reverse the current as it passes through θ=0θ=0 size 12{θ=0} {} with automatic switches called brushes. (See [link].)

Meters, such as those in analog fuel gauges on a car, are another common application of magnetic torque on a current-carrying loop. [link] shows that a meter is very similar in construction to a motor. The meter in the figure has its magnets shaped to limit the effect of θθ size 12{θ} {} by making BB size 12{B} {} perpendicular to the loop over a large angular range. Thus the torque is proportional to II size 12{I} {} and not θθ size 12{θ} {}. A linear spring exerts a counter-torque that balances the current-produced torque. This makes the needle deflection proportional to II size 12{I} {}. If an exact proportionality cannot be achieved, the gauge reading can be calibrated. To produce a galvanometer for use in analog voltmeters and ammeters that have a low resistance and respond to small currents, we use a large loop area AA size 12{A} {}, high magnetic field BB size 12{B} {}, and low-resistance coils.