Force on a moving charge in uniform magnetic and electric fields:
Derivation of magnetic force on a current carrying conductor
Consider a rod of a uniform cross-sectional area A and length l. We shall assume one kind of mobile carriers as in a conductor (here electrons). Let the number density of these mobile charge carriers in it be n. Then the total number of mobile charge carriers in it is nAl. For a steady current I in this conducting rod, we may assume that each mobile carrier has an average drift velocity vd . In the presence of an external magnetic field vector B, the force on these carriers is:
where q is the value of the charge on a carrier. Now nqvd is the current density j and |(nq vd)|A is the current I. Thus,
where vector l is a vector of magnitude l, the length of the rod, and with a direction identical to the current I.
