An electron of mass $m_e$ revolves around a nucleus of charge $+Ze$. Show that it behaves like a tiny magnetic dipole. Hence prove that the magnetic moment associated with it is $\vec{\mu} = -\dfrac{e}{2m_e}\vec{L}$, where $\vec{L}$ is the orbital angular momentum of the electron. Give the significance of the negative sign.

$\vec\mu = -\dfrac{e}{2m_e}\vec L$; negative sign: $\vec\mu$ antiparallel to $\vec L$.

CBSE Class 12 · 2017 · Delhi · Magnetic Effects of CurrentShort Answer

An electron of mass

m_{e}

revolves around a nucleus of charge

+ Z e

. Show that it behaves like a tiny magnetic dipole. Hence prove that the magnetic moment associated with it is

μ = - \frac{e}{2 m _{e}} L

, where

L

is the orbital angular momentum of the electron. Give the significance of the negative sign.

Answer: $μ = - \frac{e}{2 m _{e}} L$ ; negative sign: $μ$ antiparallel to $L$ .

An electron revolving in a circular orbit constitutes a current loop and therefore behaves like a tiny magnetic dipole. The current is

I = \frac{e}{T} = \frac{e v}{2 π r} (T = \frac{2 π r}{v}) .

Magnetic moment

μ = I A = \frac{e v}{2 π r} π r^{2} = \frac{e v r}{2}

.
Multiplying and dividing by

m_{e}

and using orbital angular momentum

L = m_{e} v r

μ = \frac{e ( m _{e} v r )}{2 m _{e}} = \frac{e}{2 m _{e}} L .

Since the electron's charge is negative, the magnetic moment points opposite to

L

μ = - \frac{e}{2 m _{e}} L .

The negative sign signifies that the magnetic moment is antiparallel to the orbital angular momentum.

μ = - \frac{e}{2 m _{e}} L

CBSE Class 12 Board · 2017 · Delhi · Subjective