Question

Magnetic moment of an electron moving in a circular orbit of radius \(r\) with a speed \(v\) is

• A. \( \dfrac{ev^2}{r} \)
• B. \( evr \)
• C. \( \dfrac{ev^2}{2r} \)
• D. \( \dfrac{evr}{2} \)

Hint:

For an electron revolving in a circular orbit: \[ \mu=\frac{evr}{2} \] This is one of the most important formulas in atomic physics and magnetism.

Answer 1

The Correct Option is D

Step 1: Treat the orbit as a current loop.
An electron going round and round is like a tiny current loop. The magnetic moment of a current loop is \[ \mu = I A \] where $I$ is the loop current and $A$ is its area.

Step 2: Find the equivalent current.
The electron passes a point once every time period $T$, so the current is charge per time. \[ I = \frac{e}{T} \]
Step 3: Write the time period.
The electron covers the circumference at speed $v$. \[ T = \frac{2\pi r}{v} \] So \[ I = \frac{ev}{2\pi r} \]
Step 4: Write the orbit area.
\[ A = \pi r^2 \]
Step 5: Multiply current by area.
\[ \mu = I A = \left(\frac{ev}{2\pi r}\right)(\pi r^2) \]
Step 6: Simplify.
The $\pi$ cancels and one $r$ cancels. \[ \mu = \frac{evr}{2} \] \[ \boxed{\dfrac{evr}{2}} \]