Question
Derive an expression for the force experienced by a current carrying straight conductor placed in a magnetic field. Under what condition is this force maximum?
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Answer
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 $\theta$ 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
$\text{I}=\text{neAv}_\text{d}\ \ .....(\text{i})$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.Need a full question paper?
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