Derive an expression for the force experienced by a moving charge in a magnetic field?

A conductor consists of large no. of free electrons. Current on the conductor means drifting of such a free electron in any fixed direction due to the motion of such a free electron each electron experience magnetic force and hence conductor itself experience magnetic force. So, when a current carrying conductor is placed in a uniform magnetic field it experience force.

Consider a conductor having length l, cross-section area A in a uniform magnetic field. If n be the number of electrons per unit volume (electron density), vd be the drift velocity of electron having electronic charge e then the current on the conductor is

Due to the motion of electron on magnatic field, each electron experience Lorent’s magnatic force.

Total number of current on the conductor is

Total force experienced by the conductor is

So the force experienced by the conductor is perpendicular to the plane containing l and B.

Special Case:

we have

Case-I:

if θ = 90°

when the current carrying conductor is placed perpendicular to the magnetic field, magnetic field produces maximum force on the conductor.

Case-II:

if θ = 0° or 180°

when the current carrying conductor is placed parallel or anti parallel with magnetic field then magnetic field produces no force on the conductor.