Question

Derive an expression for the magnetic moment of an electron revolving around the nucleus in terms of its angular momentum. What is the direction of the magnetic moment of the electron with respect to its angular momentum?

Answer 1

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Answer 2

Expression for the magnetic moment of an electron in terms of its angular momentum:

Magnetic moment:

$\bold{\mu=iA}$

Where,

i = Current = electron per unit time

$\bold{i=\frac{e}{T}}$

Time is distance per unit velocity

$\bold{i=\frac{ev}{2\pi r}}$

A = Area

Since, the electron is revolving around the nucleus.

$\bold{A = \pi r^2}$

Thus, the magnetic moment is:

$\bold{\mu=\frac{ev}{2\pi r}\times \pi r^2}$

$\bold{\therefore \mu=\frac{evr}{2}}$

Angular moment:

$\bold{L=mvr}$

$\bold{\Rightarrow vr=\frac{L}{m}}$

On substituting angular moment in magnetic moment, we get,

$\bold{\overrightarrow{\mu}=\frac{-e\overrightarrow{L}}{2m}}$

Since, the charge of the electron is negative, '-' sign is introduced. So, the magnetic moment of the electron is opposite direction with respect to its angular moment.