Force on a current carrying conductor on the basis of force on a moving charge: Consider a metallic conductor of length L, cross-sectional area A placed in a uniform magnetic field B and its length makes an angle θ with the direction of magnetic field B. The current in the conductor is I.
According to free electron model of metals, the current in a metal is due to the motion of free electrons. When a conductor is placed in a magnetic field, the magnetic field exerts a force on every free-electron. The sum of forces acting on all electrons is the net force acting on the conductor. If vd is the drift velocity of free electrons, then
Current I = neAvd ...(i)
Where n is number of free electrons per unit volume.
Magnetic force on each electron = evd B sin θ ...(ii)
Its direction is perpendicular to both \(\overrightarrow{Vd}\) and \(\overrightarrow{B}\)
Volume of conductor V = AL
Therefore, the total number of free electrons in the conductor = nAL
Net magnetic force on each conductor
F = (force on one electron) × (number of electrons)
= (evdB sin θ) . (nAL) = (neAvd). BL sin θ
Using equation (i) F=IBL sin θ ...(iii)
∴ F=ILB sin θ
This is the general formula for the force acting on a current carrying conductor.
In vector form \(\overrightarrow{F}=I\overrightarrow{L}\times\overrightarrow{B}\) ....(iv)
Force will be maximum when sin θ = 1 or θ = 90°. That is when length of conductor is perpendicular to magnetic field.
