8.5 Force and Torque on a Current Loop
LEARNING OBJECTIVES
- Evaluate the net force on a current loop in an external magnetic field
- Evaluate the net torque on a current loop in an external magnetic field
- Define the magnetic dipole moment of 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 8.5.2). 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 8.5.1)
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 8.5.2 shows a rectangular loop of wire that carries a current and has sides of lengths and . The loop is in a uniform magnetic field: . The magnetic force on a straight current-carrying wire of length is given by . To find the net force on the loop, we have to apply this equation to each of the four sides. The force on side is
where the direction has been determined with the RHR-1. The current in side flows in the opposite direction to that of side , so
The currents in sides and are perpendicular to and the forces on these sides are
We can now find the net force on the loop:
Although this result 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 8.5.2)
To find the net torque on the current loop shown in Figure 8.5.2, we first consider and . 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 and . Let’s calculate the torques around the axis that passes through point of Figure 8.5.2 (a side view of the coil) and is perpendicular to the plane of the page. The point is a distance from side and a distance from side of the loop. The moment arms of and are and , respectively, so the net torque on the loop is
This simplifies to
where is the area of the loop.
Notice that this torque is independent of ; it is therefore independent of where point is located in the plane of the current loop. Consequently, the loop experiences the same torque fr