22.7 Magnetic force on a current-carrying conductor

Learning objectives

By the end of this section, you will be able to:

• Describe the effects of a magnetic force on a current-carrying conductor.
• Calculate the magnetic force on a current-carrying conductor.

The information presented in this section supports the following AP® learning objectives and science practices:

• 3.C.3.1 The student is able to use right-hand rules to analyze a situation involving a current-carrying conductor and a moving electrically charged object to determine the direction of the magnetic force exerted on the charged object due to the magnetic field created by the current-carrying conductor. (S.P. 1.4)

Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself.

We can derive an expression for the magnetic force on a current by taking a sum of the magnetic forces on individual charges. (The forces add because they are in the same direction.) The force on an individual charge moving at the drift velocity $v_{\mathrm{d}}$ is given by $F={\text{qv}}_{\mathrm{d}}B\text{sin}\theta$ . Taking $B$ to be uniform over a length of wire $l$ and zero elsewhere, the total magnetic force on the wire is then $F=({\text{qv}}_{\mathrm{d}}B\text{sin}\theta )(N)$ , where $N$ is the number of charge carriers in the section of wire of length $l$ . Now, $N=\text{nV}$ , where $n$ is the number of charge carriers per unit volume and $V$ is the volume of wire in the field. Noting that $V=\text{Al}$ , where $A$ is the cross-sectional area of the wire, then the force on the wire is $F=({\text{qv}}_{\mathrm{d}}B\text{sin}\theta )(\text{nAl})$ . Gathering terms,

Because ${\text{nqAv}}_{\mathrm{d}}=I$ (see Current ),

is the equation for magnetic force on a length $l$ of wire carrying a current $I$ in a uniform magnetic field $B$ , as shown in [link] . If we divide both sides of this expression by $l$ , we find that the magnetic force per unit length of wire in a uniform field is $\frac{F}{l}=\text{IB}\text{sin}\theta$ . The direction of this force is given by RHR-1, with the thumb in the direction of the current $I$ . Then, with the fingers in the direction of $B$ , a perpendicular to the palm points in the direction of $F$ , as in [link] .

Magnetic force on current-carrying conductors is used to convert electric energy to work. (Motors are a prime example—they employ loops of wire and are considered in the next section.) Magnetohydrodynamics (MHD) is the technical name given to a clever application where magnetic force pumps fluids without moving mechanical parts. (See [link] .)