Magnetic Fields

The study of magnetic fields stretches back millennia, from the earliest observations of lodestone attracting iron to the sophisticated electromagnetic theory that underpins contemporary physics. Ancient Greek philosophers noted that certain stones found near the city of Magnesia in modern-day Turkey could attract iron filings, and Chinese navigators exploited this phenomenon in the magnetic compass as early as the eleventh century. Yet the leap from curiosity to quantitative science required centuries of painstaking experimentation, and it was not until the nineteenth century that physicists unified electricity and magnetism into a single coherent framework. Understanding this historical trajectory reveals why magnetic fields occupy such a central position in physics — they represent a paradigm shift from the idea of action at a distance to the modern concept of a field permeating space, mediating forces between charges and currents.

This rich history raises several interconnected questions that this lesson addresses: What exactly is a magnetic field, and how do we describe it mathematically? How do magnetic fields exert forces on moving charges and current-carrying conductors? And how does the superposition principle allow us to compute the net field produced by complex current distributions? These are the foundational questions of magnetostatics, and mastering them is prerequisite to understanding electromagnetic induction, Maxwell's equations, and ultimately the nature of light.

A magnetic field, denoted B, is a vector field that assigns a magnitude and direction to every point in space, characterizing the influence that moving charges, electric currents, and magnetized materials exert on their surroundings. Unlike the electric field, which acts on charges regardless of their state of motion, the magnetic field produces a force only on charges in motion. The SI unit of magnetic field strength is the tesla (T), where 1 T = 1 kg·s⁻²·A⁻¹. The older CGS unit, the gauss (G), relates by 1 T = 10⁴ G; the Earth's surface field is roughly 25–65 μT (0.25–0.65 G). Understanding the following core principles is essential before diving into the mathematics.

A powerful way to build intuition for magnetic fields is to examine the field-line patterns produced by common sources. The diagram below illustrates the magnetic field of a bar magnet (equivalently, a magnetic dipole) alongside the field created by a long straight current-carrying wire. Notice that the bar magnet's field lines emerge from the north pole and loop back into the south pole, forming continuous closed curves. The wire's field lines, by contrast, form concentric circles centered on the wire, with the direction given by the right-hand rule: point the thumb of your right hand in the direction of conventional current, and your curled fingers indicate the circulation direction of B.

Several features of these diagrams deserve emphasis. First, the density of field lines in the bar-magnet diagram is greatest near the poles, reflecting the fact that the field magnitude is strongest there. Second, the field lines never cross — if they did, B would have two directions at that point, which is physically meaningless. Third, for the straight wire, the field magnitude decreases as 1/r, which is apparent from the increasing spacing of the concentric circles. This 1/r dependence contrasts with the 1/r² fall-off of the electric field from a point charge, and it arises because the wire is an extended, one-dimensional source rather than a zero-dimensional point source.

The mathematical description of magnetic fields in the classical regime rests on two complementary formalisms: the Lorentz force law, which specifies how a given field acts on charges and currents, and the Biot–Savart law and Ampère's law, which specify how charges and currents produce the field. Together these equations fully describe magnetostatics — the regime of steady (time-independent) currents and fields.

Applying the Biot–Savart law or Ampère's law to standard geometries yields canonical results that appear throughout physics and engineering. The table below summarizes the most important cases, and the following diagram illustrates the field inside a solenoid — one of the most useful field-producing devices because it generates a nearly uniform field over its interior volume.

The derivation of the solenoid field via Ampère's law is elegant. Choose a rectangular Amperian loop whose long side of length ℓ runs inside the solenoid parallel to the axis, and whose return path lies outside where B ≈ 0. The line integral then reduces to Bℓ = μ₀(nℓ)I, giving B = μ₀nI. This result is remarkable because it is independent of the position inside the solenoid and independent of the solenoid's radius — the field depends only on the turn density n and the current I. Real solenoids of finite length deviate from this ideal near their ends, where fringe fields emerge, but the approximation holds well over the central region.

A proton (q = 1.60 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) enters a region of uniform magnetic field B = 0.50 T directed in the +z direction. The proton's velocity is v = 2.0 × 10⁶ m/s entirely in the +x direction. Determine the radius of the circular orbit, the cyclotron frequency, and the period of revolution.

Students often find it helpful to compare the magnetic field side by side with the electric field, since many of the mathematical structures are analogous while the physical behaviors differ profoundly. The table below highlights these parallels and distinctions, which become even more meaningful once you encounter Maxwell's equations and special relativity.

The magnetostatic framework presented in this lesson is the starting point for a much deeper theory. Adding time dependence leads to Faraday's law of induction (a changing magnetic flux induces an EMF) and Maxwell's displacement current correction to Ampère's law. Together, these four Maxwell's equations unify electricity, magnetism, and optics. Special relativity further reveals that E and B are components of a single electromagnetic field tensor Fᵘᵛ, and what one observer calls a purely electric force, another observer in a different reference frame may interpret as partly magnetic.

Looking ahead, the magnetic field also plays a pivotal role in quantum mechanics — orbital and spin magnetic moments of electrons determine atomic spectra and are the basis of magnetic resonance imaging (MRI). In condensed matter physics, the response of materials to applied B fields gives rise to phenomena ranging from diamagnetism and paramagnetism to the exotic quantum states of superconductors and topological insulators. Mastering the classical magnetic field concepts in this lesson prepares you for all of these advanced topics.