CBSE 2025, Set 1

Question

An electron of mass $m$ and charge $-e$ is revolving anticlockwise around the nucleus of an atom.

(a) Obtain the expression for the magnetic dipole moment ($\mu$) of the atom.

(b) If $\vec{L}$ is the angular momentum of electron, show that $\vec{\mu} = -\left(\dfrac{e}{2m}\right)\vec{L}$.

Edvaya · Accepted Answer

The correct answer is Subjective. **(a) Expression for magnetic dipole moment:** An electron revolving in a circular orbit of radius $r$ with speed $v$ constitutes a current loop. The time period of revolution: $T = \dfrac{2\pi r}{v}$ The equivalent current (note: electron moves anticlockwise, conventional current is clockwise): $$I = \frac{e}{T} = \frac{e}{\dfrac{2\pi r}{v}} = \frac{ev}{2\pi r}$$ The magnetic dipole moment: $$\mu = I \cdot A = \frac{ev}{2\pi r} 	imes \pi r^2$$ $$\boxed{\mu = \frac{1}{2}evr}$$ **(b) Relation between $\vec{\mu}$ and $\vec{L}$:** The angular momentum of the electron: $$L = mvr \implies vr = \frac{L}{m}$$ Substituting into the expression for $\mu$: $$\mu = \frac{1}{2}e \cdot vr = \frac{1}{2}e \cdot \frac{L}{m} = \frac{eL}{2m}$$ In **vector form**, since the electron has a negative charge and revolves anticlockwise, the magnetic moment vector $\vec{\mu}$ points **opposite** to the angular momentum $\vec{L}$: $$\boxed{\vec{\mu} = -\left(\frac{e}{2m}ight)\vec{L}}$$ > **Physical significance:** The ratio $\mu/L = e/2m$ is called the **gyromagnetic ratio** of the electron. In quantum mechanics, when $\vec{L}$ is replaced by spin angular momentum, the spin magnetic moment includes an extra factor of ~2 (the Landé g-factor).

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Physics > Magnetism Induction And Alternating Current > Moving Charges And Magnetism

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