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Magnetic Dipole Moment of a Revolving Electron

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Estimated time: 5 minutes
CISCE: Class 12

Introduction

An electron revolving around the nucleus completes a closed path repeatedly. A moving charge in a closed path is equivalent to a current loop. Therefore, the electron possesses a magnetic dipole moment.

CISCE: Class 12

Derivation: Orbital Magnetic Dipole Moment

Step 1: Current due to the revolving electron

The magnitude of charge completing one revolution in time T is e. Hence, the magnitude of equivalent current is: I = \[\frac {e}{T}\]

For circular motion: T = \[\frac {2πr}{v}\]

Therefore, I = \[\frac {e}{2πr/v}\] = \[\frac {ev}{2πr}\]   (1)

Step 2: Area enclosed by the orbit

A = πr2   (2)

Step 3: Magnetic dipole moment

For a current loop: μ = IA

Using Eqs. (1) and (2):

μ = (\[\frac {ev}{2πr}\])(πr2)
μ = \[\frac {evr}{2}\]   (3)
Result: The magnitude of the orbital magnetic dipole moment is proportional to orbital speed and orbit radius.
CISCE: Class 12

Relation with Orbital Angular Momentum

The magnitude of the electron's orbital angular momentum is:

L = mevr   (4)

Dividing Eq. (3) by Eq. (4): \[\frac{\mu}{L}=\frac{evr/2}{m_evr}\]

\[{\frac{\mu}{L}=\frac{e}{2m_e}}\]   (5)

This constant ratio is called the orbital gyromagnetic ratio of the electron: γ = \[\frac {e}{2m_e}\]

Vector Form

Since an electron is negatively charged, its magnetic moment is opposite to its orbital angular momentum:

\[{\vec{\mu}=-\frac{e}{2m_e}\vec{L}}\]   (6)
Important: The negative sign indicates that \[\vec μ\]​ and \[\vec L\] are antiparallel. It does not indicate a negative magnitude.
CISCE: Class 12

Units and Dimensional Note

Quantity SI Unit Equivalent Unit
Magnetic dipole moment μ A m2 J T−1

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Related QuestionsVIEW ALL [19]

Let r be the distance of a point on the axis of a magnetic dipole from its centre. The magnetic field at such a point is proportional to

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µs = –(e/m) S,

µl = –(e/2m) l

Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

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