Q.An electron of mass $m$ and charge $-e$ is revolving anticlockwise around the nucleus of an atom.

Concept: Orbital magnetic dipole moment of an electron — a moving charge in a closed loop behaves like a current loop, producing a magnetic moment proportional to its angular momentum.

(a) The electron revolving with speed $v$ in a circular orbit of radius $r$ completes one revolution in time $T = 2\pi r / v$. This constitutes a current $I = \frac{-e}{T} = -\frac{ev}{2\pi r}$.

The magnetic dipole moment is $\mu = I \times \text{area} = I \cdot \pi r^2$. Substituting $I$:

$\mu = \left(-\frac{ev}{2\pi r}\right) \pi r^2 = -\frac{evr}{2}.$ …

The key insight here is that a moving charge creates a current loop, and any current loop has a magnetic dipole moment. When an electron orbits a nucleus, it behaves exactly like a tiny current loop — the electron's motion is equivalent to a current flowing in a circular path. The magnetic dipole moment depends on both the current (how fast the charge is moving) and the area of the loop.

The negative sign in the final expression comes from the electron's negative charge. If the electron were positively charged, the magnetic moment would point in the same direction as the angular momentum. But since it's negative, the magnetic moment points opposite to the angular momentum vector.

Let's work through this systematically.

1. Understanding the physical picture. The electron moves in a circular orbit around the nucleus. Its motion constitutes a current — every time the electron completes one revolution, it has transported one unit of charge ($-e$) past any point on the orbit. The direction of conventional current is opposite to the electron's motion (since current is defined as the flow of positive charge).

2. Finding the current. The time taken for one complete revolution is the time period $T$. If the electron's speed is $v$ and the orbital radius is $r$, then:

Current $I$ is charge flowing per unit time. The charge flowing past a point in one revolution is $e$ (magnitude), so:

This is the magnitude of the current. The direction of conventional current is opposite to the electron's motion.

3. Magnetic dipole moment of a current loop. For any planar current loop, the magnetic dipole moment $\vec{\mu}$ has magnitude $I \times A$, where $A$ is the area enclosed. The direction is given by the right-hand rule: curl your fingers in the direction of the current, and your thumb points in the direction of $\vec{\mu}$.

   The area of the circular orbit is $A = \pi r^2$. Therefore:

This is the magnitude of the magnetic dipole moment.

4. Relating to angular momentum. The electron's orbital angular momentum $\vec{L}$ has magnitude $L = mvr$ (for a particle moving in a circle, $L = r \times p = rmv$). The direction of $\vec{L}$ is given by the right-hand rule for angular momentum: curl your fingers in the direction of motion, and your thumb points along $\vec{L}$. …