Question

Obtain an expression for the orbital magnetic moment of an electron rotating about the nucleus in an atom.

Answer 1

Hint: In this question, we will use the relation between the magnetic dipole moment, current and the area. Further, by substituting the values of electron and atom, we get the required result. Also, we will discuss the basics of time, angular momentum, for our better understanding.
Formula used:
$M = IA$
$L = mvr$

Complete answer:
Let us suppose, an electron revolving around a nucleus in an atom. As we know that, the electron is a negatively charged particle, so atom will act as a current carrying loop and the magnetic dipole moment of an electron is given by:
$M = IA$
Here, I is current and A is the area of the cross section.
$M = \dfrac{e}{t}\left( {\pi {r^2}} \right)$
Now, by putting the value of time t, we get:
$M = \left( {\dfrac{e}{{\dfrac{{2\pi r}}{v}}}} \right)\left( {\pi {r^2}} \right)$
$ \Rightarrow M = \dfrac{{evr}}{2}$
Now, as we know that the angular momentum is given by:
$L = mvr$
$\therefore M = \left( {\dfrac{2}{{2{m_e}}}} \right)L$
Therefore, the magnetic dipole moment of the electron and angular momentum are in the opposite direction and we get the required result.

Additional information:
As we know, time is defined by its measurement. Time is what a clock reads. In classical and non- relativistic physics, it is a scalar quantity. We know that length, mass and charge is usually described as a fundamental quantity, similarly time is also a fundamental quantity. The S.I unit of time is second.
When electrons are emitted when electromagnetic radiation, such as light, hits the material or its surface is called the photoelectric effect. The work function for photoelectric effect is different for different materials. It depends upon the intensity of light.
Angular velocity is also known as rotational velocity. It is measured in angle per unit time or radian per second (rad/sec).
The rate of change of angular velocity is angular acceleration.
Angular momentum is the rotational equivalent to linear momentum, given by:
$L = mvr$
Here, L is linear momentum, m is mass, v is velocity and r is the radius of an object.
It is a conserved quantity. The total angular momentum of a closed system remains constant.
Angular momentum is also the product of inertia and angular velocity. If angular momentum remains constant than the angular velocity of the object must increase

Note:
We should remember that the direction of the turntable is opposite to the direction of man. Also, the initial and final linear momentum is equal when there is no torque i.e., conserved angular moment. The relation between linear moment and angular moment need to remember.