Magnetic Dipole Moment of a Revolving Electron

Introduction

The magnetic strength of the magnet, its direction and the other object responsible for producing a magnetic field are measured by magnetic moment. The magnetic moment is more accurately known as the magnetic dipole moment and is represented by a magnetic dipole. To create an image of a magnetic dipole, you can imagine a magnet with two poles – the North and the South pole, having a small distance between them.

When an electron starts revolving around the nucleus of an atom, an electric current is set up. The motion of the electron and the direction of the electric current operate opposite to each other. For instance, if the motion of an electron operates in the anticlockwise direction, the conventional current gets set up in the clockwise direction. Interested in exploring more regarding the magnetic dipole moment of a revolving electron? Keep reading on!

Definition of Magnetic Dipole Moment

An electronic magnetic moment is induced due to two major facts – electric charge and the electron’s spin. When an electron revolves around the nucleus, the magnetic dipole moment of a revolving electron is induced. It can be calculated by the formula derived below. 

Derivation of Magnetic Dipole Moment of Electron

Suppose an electron of charge e revolves around the nucleus of charge Ze. The electrons revolve around the atom in orbit, having a radius of r, velocity v, and μl is the magnetic moment. Here is how to find current with the revolution of electrons around the nucleus:

i = e/T

Where i = current, e = charge on an electron and T is the time period. But, the time period of the electron is T = 2πr/v. On substitution of the value of time period in the equation, we get:

i = ev/(2πr)

Now, we know that magnetic moment is given by:

μl = iA

Where A = area enclosed by the loop or electron, and the area of the orbital electronic path is πr2. Therefore, we substitute both the values of A and i. 

μl = ev/(2πr) × πr2 = evr/2

Now, multiply and divide the mass of electron me in the above equation, we get:

μl = emevr/(2me)

Here, mevr = L, where L is the electron’s angular momentum. The above equation now becomes:

μl = -eL/(2me)

In this case, one thing to note is the minus sign that comes in front of it. This minus sign indicates that the direction of the magnetic dipole moment is opposite to that of the angular magnetic moment. On rearranging the magnetic moment formula of the revolving electron becomes:

μl/L =  e/(2me)

The left-hand side ratio, μl/L, is the gyromagnetic ratio, which is a constant with a value of 8×1010 C/kg.

Unit of Magnetic Dipole Moment of a Revolving Electron

Unit for magnetic dipole moment in metre–kilogram–second–ampere measurement  system is Ampere-meter2. Unit for magnetic dipole moment in centimetre–gram–second measurement system is erg per gauss.

Bohr Magneton & its Formula

As discussed above, the magnetic moment formula is μl/L =  e/(2me). From here, the Bohr magneton formula can be easily derived. According to Bohr, the angular momentum of an electron has a discrete set of values. As per the equation, the value comes to be:

L = nh / 2 π

Where n = any natural number, and h is the Planck’s constant, 6.626 × 10-34 Js. On substituting the values in the given equation, we get:

μl = (e/2m)  (nh/2π) = neh / 4πm

When it comes to getting the minimum value of the magnetic dipole moment, the value of n is supposed to be 1.

μl (minimum) = eh/4πm 

In this case, eh/4πm is known as the Bohr magneton. It has all the constants in it. Therefore, it is easy to get the value of the Bohr magneton. When all the values of the h, e and m are substituted in the equation, the Bohr magneton comes as 9.27 × 10-24 Am2. The unit of the Bohr magneton is Ampere-meter2.

Conclusion

When the charge moves in any of the substances, it directly leads to the formation of current. The same goes for when it comes to explaining the electron that revolves around the nucleus. Neil Bohr’s atom model adequately explains this concept. As per the model, when the electron (negatively charged) moves in the circular orbit around the nucleus (positively charged), it induces an electric current. When the electrons revolve around the nucleus, it leads to the formation of the magnetic moment due to orbital motion and spin of electrons.