Question

The ratio of magnetic moment ${\mu _1}$ to the orbital angular momentum $\left( l \right)$ is Gyromagnetic ratio. Its numerical value for an electron can be given by –
A) $8.8 \times {10^{ - 12}}C/kg$
B) $8.8 \times {10^{10}}C/kg$
C) $1.6 \times {10^{ - 19}}C/kg$
D) $6.67 \times {10^{11}}C/kg$

Answer 1

Hint:- The Gyromagnetic ratio is the constant value. Find how the numerical value of Gyromagnetic ratio for a charged particle is given. Then, for an electron, mass of electron is equal to $9.1 \times {10^{ - 31}}kg$ and charge of an electron is equal to $1.6 \times {10^{ - 19}}C$.

Step by Step Solution: -
Gyromagnetic ratio of a particle or system can be defined as the ratio of its magnetic moment and angular momentum. It is denoted by the gamma symbol which is $\gamma$.
The S.I unit of Gyromagnetic ratio is radian per second per tesla which is equivalent to Coulomb per kilogram.
The g – factor is closely related with Gyromagnetic ratio but is different from it. The g – factor is a dimensionless quantity which is different from Gyromagnetic ratio.

Gyromagnetic ratio is sometimes also known as magnetogyric ratio.
We know that, Gyromagnetic ratio is the constant value.
When there is any charged particle then, the Gyromagnetic ratio is given by –
$\dfrac{{{\mu _1}}}{1} = \dfrac{e}{{2m}} \cdots \left( 1 \right)$
where, $e$ is the charge of the given charged particle, and
$m$ is the mass of that given charged particle
Now, according to the question, we have to calculate the Gyromagnetic ratio and the given charged particle is electron.
Therefore, let the mass of the electron be ${m_e}$ and the charge of the electron be ${e_e}$.
So, we know that –
Mass of an electron, ${m_e} = 9.1 \times {10^{ - 31}}kg$, and
Charge of an electron, ${e_e} = 1.6 \times {10^{ - 19}}C$
Putting the values of mass of an electron and charge of an electron in equation $\left( 1 \right)$, we get –
$\dfrac{{{\mu _1}}}{1} = \dfrac{{2 \times 1.6 \times {{10}^{ - 19}}}}{{9.1 \times {{10}^{31}}}}$
After doing calculations, we get –
${\mu _1} = 8.8 \times {10^{10}}C/kg$
So, the numerical value of Gyromagnetic ratio for an electron is $8.8 \times {10^{10}}C/kg$.

Hence, the correct option is (B).

Note: The value of the Gyromagnetic ratio varies by atomic species.
Spin and a particular form of the magnetic moment are collinear and directly proportional to one another connected by the constant called the Gyromagnetic ratio.