FORCE ON A CURRENT CARRYING CONDUCTOR PLACED IN A MAGNETIC FIELD

Consider a straight conductor of length L has area of cross-section A and the number of free charge Q per unit volume equal to n drifting with velocity V .

I = n A Q V

Multiplying both sides by L,

IL = n A Q V L

The direction of current flow opposite to velocity V, thus current element vector IL and V have opposite direction,

IL = ― n A Q V L

The magnetic Lorentz force having a charge Q and moving with the velocity V inside the magnetic field B is given by

f = ―Q ( V × B )

Total force on the current carrying conductor F,

F = N f

Where N is the total number of charge in the conductor , f is the force experiences each of the charge inside the conductor.

N = n × volume of conductor

N = n A L

F = n A L [ ―Q( V × B)]

= (―n A L Q V ) × B

=I L B