Magnetic Fields

Introduction

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A charge that is moving in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field.

Historical Background

The study of magnetic fields dates back to ancient times, with the earliest recorded observations of magnetism being made by the Greeks and Chinese. The term "magnet" is derived from the Greek word "magnes," referring to the region of Magnesia in Thessaly, where lodestones, naturally magnetized pieces of the mineral magnetite, were found. In the 19th century, Hans Christian Ørsted discovered that electric currents create magnetic fields, leading to the development of electromagnetism as a unified theory by James Clerk Maxwell.

Mathematical Description

Magnetic Field Vectors

The magnetic field can be represented mathematically by the magnetic field vector **B**. The direction of the magnetic field vector is defined as the direction in which the north end of a compass needle points. The magnitude of the magnetic field vector is proportional to the force exerted on a moving charge.

Maxwell's Equations

The behavior of magnetic fields is governed by Maxwell's equations, which are a set of four partial differential equations. These equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. The relevant equations for magnetic fields are:

Gauss's law for magnetism: ∇·**B** = 0
Faraday's law of induction: ∇×**E** = -∂**B**/∂t
Ampère's law (with Maxwell's correction): ∇×**B** = μ₀**J** + μ₀ε₀∂**E**/∂t

These equations imply that there are no magnetic monopoles; that is, magnetic field lines neither start nor end but form continuous loops or extend to infinity.

Sources of Magnetic Fields

Electric Currents

One of the primary sources of magnetic fields is electric currents. According to the Biot-Savart law, the magnetic field **B** produced at a point in space by a small segment of current-carrying wire is proportional to the current **I**, the length of the segment **dl**, and the sine of the angle between the segment and the line connecting the segment to the point, and inversely proportional to the square of the distance from the segment to the point.

Magnetic Dipoles

Magnetic fields can also be produced by magnetic dipoles, which are often associated with the intrinsic magnetic moments of elementary particles such as electrons. The magnetic moment **μ** of a dipole generates a magnetic field that decreases with the cube of the distance from the dipole.

Magnetic Field Lines

Magnetic field lines are a visual tool used to represent the direction and strength of a magnetic field. The density of the field lines indicates the strength of the magnetic field: the closer the lines, the stronger the field. Field lines emerge from the north pole of a magnet and enter the south pole, forming closed loops.

Applications of Magnetic Fields

Electromagnetic Devices

Magnetic fields are fundamental to the operation of many electromagnetic devices, such as electric motors, generators, transformers, and inductors. In these devices, the interaction between electric currents and magnetic fields is harnessed to convert electrical energy into mechanical energy, or vice versa.

Magnetic Resonance Imaging (MRI)

Magnetic fields are also crucial in medical imaging techniques such as Magnetic Resonance Imaging (MRI). MRI uses strong magnetic fields and radio waves to generate detailed images of the inside of the human body. The magnetic field aligns the nuclear magnetization of hydrogen atoms in the body, and radiofrequency fields are used to systematically alter the alignment of this magnetization, producing a signal that can be used to construct images.

Earth's Magnetic Field

The Earth itself acts as a giant magnet with a magnetic field that extends from its interior out into space. This geomagnetic field is generated by the motion of molten iron alloys in the Earth's outer core through a process known as the geodynamo. The Earth's magnetic field protects the planet from the solar wind, a stream of charged particles emanating from the Sun, and is responsible for phenomena such as the auroras.

Quantum Mechanical Perspective

In quantum mechanics, the magnetic field is associated with the magnetic vector potential **A**, which is related to the magnetic field by **B** = ∇×**A**. The interaction of charged particles with magnetic fields is described by the Hamiltonian, which includes a term for the magnetic potential energy. The quantum mechanical description of magnetism also involves the concept of spin, an intrinsic form of angular momentum carried by elementary particles.