Question

Show that the frequency of revolution, of a charged particle (in the X-Y plane), in a uniform magnetic field (B) is independent of its speed.

Answer 1

In this question, we will first use the equation of force exerted on a charge, moving in a magnetic field. Further, we will use this relation in frequency to find the required result. At last we will discuss the basics of magnetic field and frequency.
Formula used:
$F = qvB$
$\nu = \dfrac{1}{T} = \dfrac{v}{{2\pi r}}$

Complete answer:
As we know that the radius of a circular path traced by a charged particle in the perpendicular uniform magnetic field is given by:
$F = qvB$
$\Rightarrow F = \dfrac{{m{v^2}}}{r}$
$\Rightarrow r = \dfrac{{mv}}{{qB}}$
Now, we know that frequency of revolution is given by:
$\nu = \dfrac{1}{T} = \dfrac{v}{{2\pi r}}$
$\Rightarrow \nu = \dfrac{{v \times qB}}{{2\pi \times mv}}$
$\therefore \nu = \dfrac{{qB}}{{2\pi m}}$
We can see that this equation is independent of speed v.
Therefore, we get the required expression of frequency which is independent of velocity v.

Additional information:
As we know that a cyclotron makes use of this expression i.e., frequency which is independent of velocity, in which alternating electric field of frequency is applied to accelerate the particle.
Also, frequency is defined as the number of occurrences of a repeating event per unit of time. There are different types of frequency, and classified as temporal frequency, spatial frequency and angular frequency. The S.I unit of frequency is called hertz and it is represented by Hz. One hertz is termed as to one occurrence of a repeating event per second.
A magnetic field B can be defined as a vector field that tells the magnetic influence on moving electric charges, electric currents, and magnetized materials. Also, we should know that a charge that is moving in a magnetic field experiences a force that is perpendicular to its town velocity and to the magnetic field.

Note:
When should remember that when an electric charge is moving in a magnetic field then it experiences a sideways magnetic force and thereby gets a turn in the direction. Magnetic field is measured in Tesla, which is represented by T.