An electron is revolving with speed $\upsilon $in a circular orbit of radius$r$. Obtain the expression of gyromagnetic ratio. What is a Bohr magneton? Write its value.

Hint: For calculating Gyromagnetic ratio, we need to find the value of Magnetic dipole moment of the electron and the Angular momentum associated with the revolving electron. For Bohr magneton, we should know the value of Planck's constant and rest mass of electrons.

Formulae used:

Gyromagnetic ratio = $\dfrac{{{M}_{o}}}{{{L}_{o}}}$
Bohr magneton ${{\mu }_{B}}=\dfrac{e\hbar }{2{{m}_{e}}}$

 Complete step by step answer:
Gyromagnetic Ratio is defined as the ratio of Magnetic Dipole moment$({{M}_{o}})$and Angular Momentum associated with electron$({{L}_{o}})$.

An electron is moving in a circular orbit of radius$r$with given speed$\upsilon $. To find the period of revolution we will use the formula,

$\text{Revolution period=}\dfrac{Circumference}{velocity}$

Circumference of orbit of radius$r$=$2\Pi r$

\[\text{Period of revolution=}\dfrac{2\Pi r}{\upsilon }\]

$\text{Current circulating meanwhile I=}\dfrac{e}{T}$ where $T$is the period of revolution we calculated above.

To find the magnitude of magnetic moment associated with the revolving electron, we will use the formula ${{M}_{o}}=IA$where $I$is the current in amperes and $A$is the area covered by the revolving electron.

${{M}_{o}}=\dfrac{e\upsilon }{2\Pi r}\times \Pi {{r}^{2}}=\dfrac{e\upsilon r}{2}$

As we already know a revolving electron has some value of Angular momentum, denoted by ${{L}_{o}}$

Above equation can be written as:

${{M}_{o}}=\dfrac{e}{2{{m}_{e}}}\times {{m}_{e}}\upsilon r=\dfrac{e}{2{{m}_{e}}}\times {{L}_{o}}$where ${{m}_{e}}$is the mass of electron

Gyromagnetic ratio = $\dfrac{{{M}_{o}}}{{{L}_{o}}}=\dfrac{e}{2{{m}_{e}}}$

Bohr Magneton is a physical quantity and a constant used to express the magnetic moment of an electron which is caused by its angular momentum, either spin or orbital momentum. It is denoted by${{\mu }_{B}}$.

Expression for Bohr Magneton:

${{\mu }_{B}}=\dfrac{e\hbar }{2{{m}_{e}}}$ where $\hbar $ represents reduced Planck's constant and ${{m}_{e}}$is the rest mass of electron.

The magnetic moment associated with an electron is approximately equal to one Bohr Magneton.

Value of one Bohr Magneton is equal to$9.274\times {{10}^{-24}}\dfrac{J}{T}$.

Note:
While calculating the value of Gyromagnetic Ratio and Bohr Magneton, always remember to work in SI units. SI unit of Gyromagnetic ratio is ‘radian per second per tesla’ or ‘coulomb per kilogram’ and SI unit of Bohr magneton is ‘ampere square meter’.