Derive an expression for force acting on a current carrying conductor when placed in a uniform magnetic field.

Explanation

When a current conductor is placed in a magnetic field, then due to motion of free electrons inside the conductor, a magnetic force is applied on it

Let us consider a potion of length l and cross-sectional area A of a straight conductor carrying a current I.
Let magnetic field B in the plane of the page directed downwards makes an angle `theta` with the direction of velocity of electron.
Let n be the number of free electrons per unit volume of the conductor and `v_(d)` be the drift velocity of the eelctrons. From the relation
`F=q(vxxB)`, where e is the (negative) charge on an eelctron. If, fowever, the conductor makes an angle `theta` with the magnetic field B measured from the conductor towards the field B, then the magnitude of the force on each elelctron is given by
`F.=ev_(d)B"sin"theta`
The number of electrons in the length l of the conductor is
N = n A l
The total force F on the free electrons, and hence on the length l of the conductor is, therefore
`F.=F.xxN=(ev_(d)B"sin"theta)(n A l)`
`=(n e A v_(d))Bl"sin"theta`
But current flowing through a conductor
`I=n eAv_(d)`
`thereforeF=IBl"sin"thetaor F=I(Bxxl)`
If `theta=0^(@)or180^(@)`,
then `F=I Bl"sin"0^(@)=0` `[because"sin"0^(@)=0and"sin"180^(@)=0]`
It means a conductor placed parallel to magnetic field, experiences no force due to magnetic field.
If `theta=90^(@)`, then foece is maximum.
`F_("max")=IBl"sin"90^(@)=IBl` `[because"sin"90^(@)=1]`
It means a conductor placed perpendicular to direction of megnetic field, experience maximum force by the magnetic force.