Question

An electron having mass $m$ and kinetic energy $E$ enter in the uniform magnetic field $B$ , perpendicularly, then its frequency will be:
a) $\dfrac{{eE}}{{qVB}}$
b) $\dfrac{{2\pi m}}{{eB}}$
c) $\dfrac{{eB}}{{2\pi m}}$
d) $\dfrac{{2m}}{{eBE}}$

Answer 1

Hint: Uniform magnetic field is the magnetic force experienced by an object when magnetic lines are parallel. Frequency is something that occurs in a unit time.it is given by the formula: $F = \dfrac{1}{T}$ , where $F = $ frequency, $T = $ time.

Complete step by step answer:
Uniform magnetic field: it is defined as the magnetic force that is experienced by an object when the magnetic lines are parallel and all are at all the same points.
For example: the strength of the bar magnet is greater towards the end of poles.
Velocity is defined as the speed of a thing at a particular direction.
The magnetic force is given by the formula: $F = qVB$
Where, $F = $ magnetic force
$q = $ charge
$V = $ particle velocity
$B = $ magnetic field
The centripetal force is given by the formula: $qVB = \dfrac{{m{v^2}}}{r}$
Where, $m = $ mass
$v = $ velocity
$r = $ radius.
So, if the velocity of the particle is perpendicular to the magnetic field, then the particle moves in a circular motion.
Then the centripetal force that is experienced by an electron will be given as $qVB$
Applying this, we will get $qVB = \dfrac{{m{v^2}}}{r}$…..1
$qB = \dfrac{{m{v^{}}}}{r}$ ….2
Now as we know, $v = \omega R$
Substituting this value, in the above equation 2 we get,
$qB = m\omega $
Therefore, $\omega = \dfrac{{qB}}{m}$…3
We know that, $\omega = \dfrac{{2\pi }}{T}$ , Substitute this value in equation 3 we get,
$\dfrac{{qB}}{M} = \dfrac{{2\pi }}{T}$ ….4
After all this we will see about the frequency.
It is defined as the number of occurrences per unit time. It is given by the formula $F = \dfrac{1}{T}$
But the value of $\dfrac{1}{T} = \dfrac{{qB}}{{2\pi m}}$
Substituting this value in equation 4 we get,
 $\dfrac{1}{T} = \dfrac{{eB}}{{2\pi m}}$
Therefore, $F = \dfrac{{eB}}{{2\pi m}}$
Where, $F = $ frequency
$e = $ energy
$B = $ magnetic field
$m = $ mass of the particle

So, the correct answer is option C) $\dfrac{{eB}}{{2\pi m}}$

Note: Magnetic force is perpendicular to the direction it travels. Therefore, the charged particles follow a curved path in the magnetic field. The magnetic field does not depend on the velocity and the radius of the circular path.