11.5 Force and Torque on a Current Loop

Motors are the most common application of magnetic force on current-carrying wires. Motors contain 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 into mechanical work in the process. Once the loop’s surface area is aligned with the magnetic field, the direction of current is reversed, so there is a continual torque on the loop (Figure 11.15). This reversal of the current is done with commutators and brushes. The commutator is set to reverse the current flow at set points to keep continual motion in the motor. A basic commutator has three contact areas to avoid and dead spots where the loop would have zero instantaneous torque at that point. The brushes press against the commutator, creating electrical contact between parts of the commutator during the spinning motion.

Figure 11.15 A simplified version of a dc electric motor. (a) The rectangular wire loop is placed in a magnetic field. The forces on the wires closest to the magnetic poles (N and S) are opposite in direction as determined by the right-hand rule-1. Therefore, the loop has a net torque and rotates to the position shown in (b). (b) The brushes now touch the commutator segments so that no current flows through the loop. No torque acts on the loop, but the loop continues to spin from the initial velocity given to it in part (a). By the time the loop flips over, current flows through the wires again but now in the opposite direction, and the process repeats as in part (a). This causes continual rotation of the loop.

In a uniform magnetic field, a current-carrying loop of wire, such as a loop in a motor, experiences both forces and torques on the loop. Figure 11.16 shows a rectangular loop of wire that carries a current I and has sides of lengths a and b. The loop is in a uniform magnetic field: $\overrightarrow{B}=B\widehat{j}.$ The magnetic force on a straight current-carrying wire of length l is given by $I\overrightarrow{l}\times \overrightarrow{B}.$ To find the net force on the loop, we have to apply this equation to each of the four sides. The force on side 1 is

where the direction has been determined with the RHR-1. The current in side 3 flows in the opposite direction to that of side 1, so

The currents in sides 2 and 4 are perpendicular to $\overrightarrow{B}$ and the forces on these sides are

We can now find the net force on the loop:

Although this result $\left(\text{Σ}F=0\right)$ has been obtained for a rectangular loop, it is far more general and holds for current-carrying loops of arbitrary shapes; that is, there is no net force on a current loop in a uniform magnetic field.

Figure 11.16 (a) A rectangular current loop in a uniform magnetic field is subjected to a net torque but not a net force. (b) A side view of the coil.

To find the net torque on the current loop shown in Figure 11.16, we first consider $F_1$ and $F_3.$ Since they have the same line of action and are equal and opposite, the sum of their torques about any axis is zero (see Fixed-Axis Rotation). Thus, if there is any torque on the loop, it must be furnished by $F_2$ and $F_4.$ Let’s calculate the torques around the axis that passes through point O of Figure 11.16 (a side view of the coil) and is perpendicular to the plane of the page. The point O is a distance x from side 2 and a distance $\left(a-x\right)$ from side 4 of the loop. The moment arms of $F_2$ and $F_4$ are $x\text{sin}\theta$ and $\left(a-x\right)\text{sin}\theta ,$ respectively, so the net torque on the loop is

This simplifies to

where $A=ab$ is the area of the loop.

Notice that this torque is independent of x; it is therefore independent of where point O is located in the plane of the current loop. Consequently, the loop experiences the same torque from the magnetic field about any axis in the plane of the loop and parallel to the x-axis.

A closed-current loop is commonly referred to as a magnetic dipole and the term IA is known as its magnetic dipole moment $\mu .$ Actually, the magnetic dipole moment is a vector that is defined as

where $\widehat{n}$ is a unit vector directed perpendicular to the plane of the loop (see Figure 11.16). The direction of $\widehat{n}$ is obtained with the RHR-2—if you curl the fingers of your right hand in the direction of current flow in the loop, then your thumb points along $\widehat{n}.$ If the loop contains N turns of wire, then its magnetic dipole moment is given by

In terms of the magnetic dipole moment, the torque on a current loop due to a uniform magnetic field can be written simply as

This equation holds for a current loop in a two-dimensional plane of arbitrary shape.

Using a calculation analogous to that found in Capacitance for an electric dipole, the potential energy of a magnetic dipole is