11.5 Force and torque on a current loop  (Page 2/5)

To find the net torque on the current loop shown in [link] , 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 [link] (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 [link] ). 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

Summary

• The net force on a current-carrying loop of any plane shape in a uniform magnetic field is zero.
• The net torque τ on a current-carrying loop of any shape in a uniform magnetic field is calculated using $\tau =\overrightarrow{\mu }\times \overrightarrow{B}$ where $\overrightarrow{\mu }$ is the magnetic dipole moment and $\overrightarrow{B}$ is the magnetic field strength.
• The magnetic dipole moment $\mu$ is the product of the number of turns of wire N , the current in the loop I , and the area of the loop A or $\overrightarrow{\mu }=NIA\widehat{n}.$

Problems

(a) A 200-turn circular loop of radius 50.0 cm is vertical, with its axis on an east-west line. A current of 100 A circulates clockwise in the loop when viewed from the east. Earth’s field here is due north, parallel to the ground, with a strength of $3.0\times {10}^{\mathrm{-5}}\text{T}.$ What are the direction and magnitude of the torque on the loop? (b) Does this device have any practical applications as a motor?

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Repeat the previous problem, but with the loop lying flat on the ground with its current circulating counterclockwise (when viewed from above) in a location where Earth’s field is north, but at an angle 45.0° below the horizontal and with a strength of $6.0\times {10}^{\mathrm{-5}}\text{T}.$

$0.666\text{N}·\text{m}$

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