Force on a Current-Carrying Conductor in a Magnetic Field

From individual charge carriers to the whole conductor. A current-carrying conductor

contains a very large number of free charge carriers (electrons, in a metal), each drifting with

some small average velocity $\vec{v}_d$ (the drift velocity) along the wire. If the conductor sits in

an external magnetic field $\vec{B}$, EVERY one of these moving charge carriers individually feels

the magnetic Lorentz force of Section 4.8, $\vec{f} = q\vec{v}_d\times\vec{B}$; the conductor as a

whole then feels the combined, summed effect of every one of these microscopic forces -- a genuine,

measurable, macroscopic force on the wire.

Deriving $F=BIL\sin\theta$. Consider a straight segment of conductor of length $L$ and

cross-sectional area $A$, carrying current $I$, with $n$ free charge carriers (each of charge $q$)

per unit volume. The number of charge carriers in this segment is $N = n(AL)$, and the current

itself is related to the drift velocity by the standard relation $I = nqv_dA$. The total magnetic

force is the sum of the force on all $N$ carriers:

and substituting $I=nqv_dA$ for the bracketed factor gives directly

where $\theta$ is the angle between the direction of current flow (i.e. the direction of $L$, taken

as a vector $\vec{L}$ along the wire) and the field $\vec{B}$. In full vector form, this is written

$\vec{F} = I\,\vec{L}\times\vec{B}$.

Special cases and direction. When the conductor is placed exactly PERPENDICULAR to the field

($\theta=90^\circ$), the force is at its maximum, $F=BIL$. When the conductor is placed exactly

PARALLEL (or antiparallel) to the field ($\theta=0^\circ$ or $180^\circ$), the force is exactly

zero -- a current-carrying wire laid along the field lines feels no magnetic force at all, for

precisely the same reason a charge moving along $\vec{B}$ feels none in Section 4.8. The direction

of the force, for the general case, is found from the vector product $\vec{L}\times\vec{B}$ using

the same right-hand rule used throughout the chapter -- or, equivalently, by the commonly taught

Fleming's left-hand rule: with the thumb, first finger, and second finger of the left hand held

mutually perpendicular, the First finger points along the Field, the seCond finger along the

Current, and the thuMb then gives the direction of the resulting Motion (force). …