Assume that an electron of mass m and charge magnitude e...

Assume that an electron of mass $m$ and charge magnitude $e$ moves in a circular orbit of radius $r$ about a nucleus. A uniform magnetic field $\vec{B}$ is then established perpendicular to the plane of the orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field $\vec{B}$ is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field.

Key Concepts

Magnetic Dipole Moment

This concept refers to the magnetic field produced by a moving charge in a closed loop. In classical physics, an electron in a circular orbit behaves like a current loop, and its magnetic dipole moment can be calculated as the product of the current and the area of the loop. This relationship is foundational in understanding how microscopic magnetic moments arise from charged particles in motion.

Lorentz Force

The Lorentz force is the force experienced by a moving charge in the presence of electric and magnetic fields. In the context of an electron orbiting in a magnetic field, the Lorentz force alters the electron's velocity, thereby modifying the current and, consequently, the magnetic dipole moment. This force is perpendicular to both the velocity of the electron and the magnetic field, and it plays a key role in the dynamics of charged particles in external fields.

Perturbation in Orbital Dynamics

When a uniform magnetic field is applied to an electron in a fixed-radius circular orbit, the resulting change in the speed of the electron is typically small. This leads to a perturbative change in the magnetic dipole moment. The perturbation analysis involves considering small corrections to the electron's motion, allowing one to linearize the equations and determine how the magnetic dipole moment is altered by the applied field.

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