Question

What is cyclotron frequency of an electron with an energy of 100 eV in the earth's magnetic field of $ 1 \times {10^{ - 4}}{\text{weber }}{{\text{m}}^{ - 2}} $ if its velocity is perpendicular to magnetic field?
(A) 0.7 MHz
(B) 2.1 MHz
(C) 2.8 MHz
(D) 1.4 MHz

Answer 1

Hint: The interaction of external forces with charged particles in a magnetic field, which are already travelling in a circular route, is referred to as cyclotron resonance. It is named after the cyclotron, a cyclic particle accelerator that adds kinetic energy to charged particles using an oscillating electric field adjusted to this resonance.

Complete answer:
The frequency of a charged particle travelling perpendicular to the direction of a uniform magnetic field B is known as the cyclotron frequency or gyrofrequency (constant magnitude and direction). Because the motion is constantly circular, the cyclotron frequency is determined by the centripetal and magnetic Lorentz forces being equal.
 $ f = \dfrac{\omega }{{2\pi }} = \dfrac{{qB}}{{2\pi m}} $
The cyclotron frequency is independent of the radius and velocity, and therefore of the particle's kinetic energy; all particles with the same charge-to-mass ratio rotate with the same frequency around magnetic field lines. This is only true in the non-relativistic limit, and it is the cyclotron's operating principle. The cyclotron frequency is also helpful in non-uniform magnetic fields, when the movement is nearly helical (assuming gradual variations in the size of the magnetic field) and uniform in the direction parallel to the magnetic field.
Now $ f = \dfrac{{qB}}{{2\pi m}} $
We substitute the values to get
Mass of electron = $ 9.1 \times {10^{ - 31}} $
B = $ 1 \times {10^{ - 4}}{\text{weber }}{{\text{m}}^{ - 2}} $
 $ f = \dfrac{{100 \times 1.6 \times {{10}^{ - 19}} \times {{10}^{ - 4}}}}{{2\pi \times 9.1 \times {{10}^{ - 31}}}} $
Hence
f = 2.8 MHz.

Note:
A cyclotron is a type of tiny particle accelerator that generates radioactive isotopes for imaging purposes. Stable, non-radioactive isotopes are fed into the cyclotron, which uses a magnetic field to accelerate charged particles (protons) to high energies. It's an electrically driven equipment that generates a beam of charged particles for medicinal, industrial, and scientific purposes. A cyclotron accelerates charged particles along a spiral route, rather than a straight line, as the name implies. This allows for a considerably longer acceleration path than a straight line accelerator.