Question

State Amperes circuital law. Use this law to find magnetic field due to straight infinite current carrying wire. How are the magnetic field lines different from the electrostatic field lines?
OR
State the principle of a cyclotron. Show that the time period of revolution of particles in a cyclotron is independent of their speeds. Why is this property necessary for the operation of a cyclotron?

Answer 1

Ampere's circuital law states that the line integral of magnetic field over a closed loop is ${\mu }_0$ times the total current passing through the loop.

That is $\oint \overrightarrow{B}.d\overrightarrow{l}={\mu }_{0}I$
$⟹B\left(2\pi R\right)={\mu }_{0}I$

$⟹B=\frac{{\mu }_{0}I}{2\pi R}$

The magnetic field lines form continuous closed loops, where as electrostatic field lines never forms closed loop.

Principle of Cyclotron- The frequency of revolution of the charged particle in magnetic field is independent of its energy. If particle is performing circular motion due to magnetic force then centripetal force = Magnetic force.

That is, $\frac{m{v}^2}{R}=qvBsin{90}^{\circ }$

$⟹R=\frac{mv}{qB}$

Time period=$\frac{2\pi R}{v}=\frac{2\pi m}{qB}$

The time period of revolution of particles in a cyclotron is independent of their speeds This property is necessary for the operation of a cyclotron to attain the resonance condition$v_a=v_c$ where ${\nu }_a$ is the frequency of the applied voltage and ${\nu }_c$ is cyclotron frequency.