Question

The gyromagnetic ratio of an electron = _________ specific charge of an electron?
${\text{A}}{\text{. 1}} \\\ {\text{B}}{\text{. 2}} \\\ {\text{C}}{\text{. }}\dfrac{1}{2} \\\ {\text{D}}{\text{. 4}} \\\$

Answer 1

In order to find the ratio, we use the individual formulae of both gyromagnetic ratio and specific charge of an electron. We divide their formula to obtain a number, which tells us how many times it is the other value.

Complete step by step answer:
The gyromagnetic ratio of an electron is defined as the ratio of magnetic momentum of an electron to its angular momentum. It is also known as magnetogyric ratio.
It is denoted by the symbol gamma ‘γ’.
It is given by the formula, $\gamma = \dfrac{{\text{q}}}{{2{\text{m}}}}$, where q is the charge of the electron and m is the mass of the electron.
The specific charge of an electron is defined as the ratio of the charge of an electron to its mass. It is a negative value as the charge of an electron is negative.
It is given by the formula specific charge = $\dfrac{{\text{q}}}{{\text{m}}}$.
Now to find the ratio of the gyromagnetic ratio and the specific charge of an electron, we divide them
$\Rightarrow \dfrac{{\dfrac{{\text{q}}}{{2{\text{m}}}}}}{{\dfrac{{\text{q}}}{{\text{m}}}}} \\\ \Rightarrow \dfrac{1}{2} \\\$
Hence the gyromagnetic ratio of an electron is one half times of the specific charge of an electron.
Option C is the correct answer

Note:
In order to answer this type of question the key is to know the definitions of the given and their formula.
The numerical value of gyromagnetic ratio of an electron is $1.76085963023{\text{ }} \times {\text{ 1}}{{\text{0}}^{11}}{\text{ }}{{\text{s}}^{ - 1}}{{\text{T}}^{ - 1}}$.
The numerical value of specific charge of an electron is $- 1.75882001076{\text{ }} \times {\text{ 1}}{{\text{0}}^{11}}{\text{ C}}{\text{.K}}{{\text{g}}^{ - 1}}$.