Question

An electron moving in a circular orbit of radius $r$ makes $n$ rotations per second. The magnetic field produced at the centre has magnitude.
$A.\dfrac{{{\mu }_{o}}{{n}^{2}}e}{r}$
$B.\dfrac{{{\mu }_{O}}ne}{2r}$
$C.\dfrac{{{\mu }_{o}}ne}{2\pi r}$
$D.$ Zero

Answer 1

We should apply our concept of Biot-Savart’s law to solve this problem. Biot-Savart’s law gives the magnetic field due to a current carrying conductor. We should start our solution using the definition of electric current. Flow of charge per unit time is called electric current. We should also consider the fact that charge associated with an electron is given as $e$.

Formula used:
We are going to solve the above given problem by the use of following given formula:-
$B=\dfrac{{{\mu }_{o}}I}{2R}$.

Complete step-by-step answer:
Suppose an electron of charge $e$ is moving on a circular orbit of radius $r$ makes $n$ rotations per second. Then the current produced, $I$ will be given as follows:-
$I=\dfrac{q}{t}$
For electron it becomes,
$I=\dfrac{e}{t}$ ………………… $(i)$
As the electron is moving on a circular path of radius $r$ for $n$ number of rotation for one second then equation $(i)$ becomes,
$I=\dfrac{ne}{1}$
$I=ne$ ……………… $(ii)$ (current is directly proportional to the number of turns)
Now, applying Biot-Savart’s law we get,
$B=\dfrac{{{\mu }_{o}}I}{2R}$ ……………………. $(iii)$
Where, $I$ is current, ${{\mu }_{o}}$ is magnetic constant and $R$ is the distance.
For this problem distance is the radius of the circle. Hence, $R$ is replaced with $r$. So, equation $(iii)$ becomes
$B=\dfrac{{{\mu }_{o}}I}{2r}$ ……………… $(iv)$
Putting value of $I$ from equation $(ii)$ in $(iv)$ we get,
$B=\dfrac{{{\mu }_{o}}ne}{2r}$.
Therefore, option $(B)$ is the correct option.

So, the correct answer is “Option B”.

Additional Information: Magnetic field is the space around the magnet in which its effect can be felt. Magnetic field has its SI unit as Tesla $(T)$. It is a vector quantity. Source of the magnetic field is the current element. Magnetic fields obey the principle of superposition.

Note: In solving these types of problems the first thing which should be taken care of is the difference between magnetic and electric fields. If we have to find the direction of the magnetic field then we will use the right hand thumb rule. Charge particles at rest do not experience force due to magnetic fields. It should also be noted that unlike electric fields, the magnetic field does not change the kinetic energy of charge particles. We should use formula and geometric dimensions correctly.