The equivalent magnetic moment of an electron revolving around a nucleus

An electron in an atom revolves around nucleus in an orbit of radius 0.5 Å. Calculate the equivalent magnetic moment, if the frequency of revolution of electron is 10¹⁰MHz

Given

Radius of orbit                =       r        =       0.5 Å          =   0.5 X 10^-10 m

Frequency of revolution =       f        =       10¹⁰ MHz   =  10¹⁶ Hz

Charge on electron         =       e        =       1.6 X 10 ^-19 C

Magnetic moment          =       M      =       ?

The electron revolving in a circular orbit around the nucleus behaves as a current loop and hence posses a magnetic momentum

The magnetic momentum can be given as

M = I A

_ _ _ _ _ _ _ _ _ _ (1)

Where

I is a current flowing through the circular loop

i.e.

I = e / T

T is a time period=1 / f

Therefore  

1 / T =  f

I = e f

_ _ _ _ _ _ _ _ _ _ (2)

And

A is a area of circular loop

i.e.

A = π r²

_ _ _ _ _ _ _ _ _ _ (3)

From equation (1) (2) and (3) we get

                            M = e f  π r²   

                            M = 1.6 X 10 ^-19 X 10¹⁶ X 3.14 X (0.5 X 10^-10 )²

                                           M = 1.6 X 10 ^-19 X 10¹⁶ X 3.14 X 0.25 X 10^-20 

                                           M = 1.6 X 3.14 X 0.25 X 10 ^-19 X 10¹⁶ X 10^-20

                                           M = 1.256 X 10 ^-23 Am²

The equivalent magnetic moment of an electron revolving around a nucleus in an orbit of radius 0.5 Å with frequency 10¹⁰MHz is 1.256 X 10 ^-23 Am²