Moving Charges and Magnetism — Physics STD 12 Science — Question

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 $\text{I}=\text{neAv}_\text{d}\ \ .....(\text{i})$

Where n is number of free electrons per unit volume.

Magnetic force on each electron $=\text{ev}_\text{d}\text{B}\sin\theta\ \ .....(\text{ii}) $

Its direction is perpendicular to both $\overrightarrow{\text{vd}}$ and $\vec{\text{B}}$

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)

$=\text{ev}_\text{d}\text{B}\sin\theta.(\text{nAL})=(\text{neAv}_\text{d}).\text{BL}\sin\theta$  $$

Using equation (i) $\text{F}=\text{IBL}\sin\theta$$$ ...(iii)

$\therefore\text{F}=\text{IBL}\sin\theta$

This is the general formula for the force acting on a current carrying conductor.

In vector form $\vec{\text{F}}=\text{I}\vec{L}\times\vec{\text{B}}$

Force will be maximum when $\sin\theta=1$ or $\theta=90^\circ$. That is when length of conductor is perpendicular to magnetic field.