Q.Deduce an expression for the magnetic dipole moment of an electron orbiting around the central nucleus.

Intuition First

Imagine a tiny charged ball spinning on its axis. Because it's spinning, it has angular momentum — the same kind of quantity that keeps a bicycle wheel stable when you're riding. But because it's charged, that spinning motion also makes it behave like a tiny loop of current, which means it has a magnetic moment — it acts like a miniature bar magnet.

So one physical motion (spinning) gives rise to two different properties: a mechanical one (angular momentum) and a magnetic one (magnetic moment). The gyromagnetic ratio is simply the number that tells you how strongly these two are linked. It answers the question: For a given amount of angular momentum, how much magnetic moment do I get?

The Precise Statement

For an electron moving in a circular orbit (like in the Bohr model of the atom), the orbital angular momentum $L$ and the magnetic moment $\mu$ are related by:

where $\gamma$ is the gyromagnetic ratio. For orbital motion of an electron, the value is:

Here $e$ is the magnitude of the electron's charge and $m_e$ is its mass.

Where Does This Come From?

Consider an electron moving in a circle of radius $r$ with speed $v$. Its orbital angular momentum is:

The electron going around once per period $T = 2\pi r / v$ is equivalent to a current $I = e/T = ev/(2\pi r)$. The magnetic moment of a current loop is current times area:

Now compare $\mu$ and $L$:

That's it. The $v$ and $r$ cancel completely — the ratio depends only on the charge and mass of the electron, not on the size or speed of the orbit.

Why This Matters

The gyromagnetic ratio is the bridge between two worlds. When you put an electron in a magnetic field, the field exerts a torque on its magnetic moment. That torque changes the angular momentum. The gyromagnetic ratio tells you exactly how fast that change happens — it determines the Larmor precession frequency:

This is the frequency at which the magnetic moment (and hence the angular momentum vector) precesses around the magnetic field direction. It's the fundamental principle behind magnetic resonance imaging (MRI) and electron spin resonance (ESR) spectroscopy. …