Magnetic Forces on Current Carrying Wire
1. The Current
The current in a wire is the charge per unit time passing a given point. In practice, it is ordinarily the negatively charged electrons that do the moving—in the direction opposite to the electric current.
λ traveling down a wire at speed v constitutes a current
I = λv,
because a segment of length vΔt, carrying charge λvΔt, passes point P in a time interval Δt. Current is actually a vector:
I = λv. [Eq.1]
Because the path of the flow is dictated by the shape of the wire, one doesn’t ordinarily bother to display the direction of I explicitly, but when it comes to surface and volume currents we cannot afford to be so casual, and for the sake of notational consistency it is a good idea to acknowledge the vectorial character of currents right from the start. A neutral wire, of course, contains as many stationary positive charges as mobile negative ones. The former do not contribute to the current—the charge density λ in Eq. 1 refers only to the moving charges. In the unusual situation where both types move, I = λ+v+ + λ−v−.
2. Magnetic Force on a current carrying wire
The magnetic force on a segment of current-carrying wire is
Fmag = ∫(v × B) dq = ∫(v × B)λ dl = ∫(I × B) dl.
Inasmuch as I and dl both point in the same direction, we can just as well write this as
Fmag = ∫I(dl × B).
Typically, the current is constant (in magnitude) along the wire, and in that case I comes outside the integral:
Fmag = I∫(dl × B).