Magnetic moment of a revolving electron of charge $e$ and mass $m$ in terms of angular momentum $L$ of the electron is:
Show Hint
The ratio of magnetic moment to angular momentum ($\frac{\mu}{L} = \frac{e}{2m}$) is a universal value known as the gyromagnetic ratio for an electron. Memorizing this constant ratio allows you to instantly solve for either variable on entrance exams.
Step 1: What we want.
An electron of charge $e$ and mass $m$ goes round in a circle. We want its magnetic moment written using its angular momentum $L$.
Step 2: Magnetic moment of a loop.
A circulating charge acts like a tiny current loop. Its magnetic moment is
\[ \mu = I \cdot A \]
the current times the area enclosed.
Step 3: Find the current.
The electron goes once round in time $T = \dfrac{2\pi r}{v}$. So the current is
\[ I = \frac{e}{T} = \frac{ev}{2\pi r} \]
Step 4: Put in the area.
The loop area is $A = \pi r^2$. So
\[ \mu = \frac{ev}{2\pi r} \times \pi r^2 = \frac{evr}{2} \]
Step 5: Bring in angular momentum.
Angular momentum is $L = mvr$, so $vr = \dfrac{L}{m}$. Substitute:
\[ \mu = \frac{e}{2} \times \frac{L}{m} \]
Step 6: Final form.
\[ \mu = \frac{eL}{2m} \]
This is option (3).
\[ \boxed{\mu = \frac{eL}{2m}} \]