Question

Dimensions of Gyromagnetic ratio are?
\[\begin{align}
  & A.[{{L}^{1}}{{M}^{0}}{{T}^{1}}{{A}^{1}}] \\
 & B.[{{L}^{0}}{{M}^{-1}}{{T}^{1}}{{A}^{1}}] \\
 & C.[{{L}^{1}}{{M}^{0}}{{T}^{0}}{{A}^{-1}}] \\
 & D.[{{L}^{-1}}{{M}^{0}}{{T}^{1}}{{A}^{1}}] \\
\end{align}\]

Answer 1

Hint: Gyromagnetic ratio is a constant and has a value $8.8\times {{10}^{10}}Ck{{g}^{-1}}$ and it is the ratio of magnetic moment of a revolving electron to its angular momentum. Dimensions of gyromagnetic ratio can be found from the units i.e. $Ck{{g}^{-1}}$.

Complete step by step answer:
Gyromagnetic ratio: Gyromagnetic ratio is defined as $\dfrac{m}{L}=\dfrac{e}{2{{m}_{e}}}$. When we put the value of charge of electron and mass of electron in the above equation. We get a constant value equals $8.8\times {{10}^{10}}Ck{{g}^{-1}}$.
To determine the gyromagnetic ratio, we have to know where this ratio comes from.
Consider, an electron of charge \[e\] performs uniform circular motion around a stationary heavy nucleus of charge \[+Ze\].
Where \[Z\] is the atomic number of that atom.
Let \[r\] be the orbital radius of electron and \[v\] be its orbital velocity then,
\[\begin{align}
  & T=\dfrac{2\pi r}{v} \\
 & I=\dfrac{e}{t} \\
 & I=\dfrac{ev}{2\pi r} \\
\end{align}\]
Now, the magnetic moment due to this electron be \[m\] then,
\[\begin{align}
  & m=current\times area \\
 & m=I\times A \\
 & m=I(\pi {{r}^{2}}) \\
 & m=\dfrac{ev}{2\pi r}(\pi {{r}^{2}}) \\
 & m=\dfrac{evr}{2} \\
\end{align}\]
Now, on dividing and multiplying the above equation by mass of electron \[({{m}_{e}})\] we get,
\[\begin{align}
  & m=\dfrac{e{{m}_{e}}vr}{2{{m}_{e}}} \\
 & m=\dfrac{eL}{2{{m}_{e}}} \\
\end{align}\]
Where:
\[L=\] Angular momentum
\[\dfrac{m}{L}=\dfrac{e}{2{{m}_{e}}}\]
This quantity is a constant and is called the Gyromagnetic ratio and its value is equals to \[8.8\times {{10}^{10}}Ck{{g}^{-1}}\]
Hence, we can also say that the Gyromagnetic ratio also equals to magnetic moment per unit angular momentum.
By the above result we are able to calculate dimensional formula of gyromagnetic ratio:
Gyromagnetic ratio \[=\dfrac{e}{2{{m}_{e}}}\]
We know that:
Current is equals to charge per unit time or mathematically,
\[I=\dfrac{e}{t}\]
Or
\[e=It\]
Therefore the Gyromagnetic ratio \[=\dfrac{It}{2{{m}_{e}}}\]
And we know that the dimensions of each quantity present in the above formula is represented by:
\[\begin{align}
  & I=[A] \\
 & t=[T] \\
 & m=[M] \\
\end{align}\]
Putting all these dimensions in the above formula we get the dimensional formula of Gyromagnetic ratio.
Dimensional formula of Gyromagnetic Ratio :
\[\begin{align}
  & =\dfrac{It}{2me} \\
 & =\dfrac{[A][T]}{[M]} \\
 & =[{{M}^{-1}}{{T}^{1}}{{A}^{1}}] \\
 & =[{{L}^{0}}{{M}^{-1}}{{T}^{1}}{{A}^{1}}] \\
\end{align}\]
Hence, the correct answer is option B.

Note: Most of the students get confused that there is no dimensional formula for a constant quantity or the quantity that are in ratio. Therefore it is suggested that before going for any result just make sure to check its formula. Because every time the formula gives us the right answer.