Straight Wire

Consider a straight wire of length $L$ (Fig. 10.8), carrying a steady current $I$, placed in an external magnetic field $\vec{B}$ applied perpendicular to the wire. Let $\vec{v}_d$ be the (common) drift velocity of the conduction electrons inside the wire. In a time $t$, the total charge $q$ that flows across any cross-section of the wire is, by the definition of current, $q=It$; and since the electrons travel with drift speed $v_d$, this same charge can also be related to the length of wire the electrons traverse in that time, giving

(using $t=L/v_d$, the time for an electron to drift the full length $L$ of the segment). Now apply the single-charge Lorentz force law (Section 10.2) to this "lumped" charge $q$, treated as moving with the drift velocity $v_d$ through the field $B$:

where the $90^\circ$ reflects the given condition that $B$ is applied perpendicular to the wire, and the two factors of $v_d$ conveniently cancel -- the DRIFT SPEED itself drops out of the final formula entirely, leaving the force expressed purely in terms of the macroscopic, directly-measurable current $I$. This is therefore the magnetic force on a straight segment of wire, of length $L$, carrying current $I$, in a field $B$ applied exactly perpendicular to the wire:

More generally, if $\vec{B}$ is NOT necessarily perpendicular to the wire, the same derivation (carried through with the general angle $\theta$ between the current direction and $\vec{B}$) gives the fully vectorial version of this force law. Defining a length vector $\vec{L}$ directed ALONG the wire, in the sense of the current flow, with magnitude equal to the wire's length:

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