Magnetic Field: Straight Wire — Compute magnetic field from long straight wires (Biot–Savart/Ampère)

The relationship between electricity and magnetism was unknown before the nineteenth century; electric currents and magnetic compasses were studied as completely separate phenomena. The pivotal moment came in 1820 when the Danish physicist Hans Christian Ørsted noticed that a compass needle deflected when placed near a wire carrying an electric current, revealing that moving charges generate magnetic fields. Within months of this startling discovery, French physicist André-Marie Ampère formulated a quantitative law relating the circulation of the magnetic field to the enclosed current, while Jean-Baptiste Biot and Félix Savart independently derived a law that gives the magnetic field contribution from an infinitesimal current element. Together, these two approaches — the Biot–Savart law and Ampère's law — form the twin pillars for computing magnetic fields from steady currents, and they remain indispensable tools in electromagnetism courses and engineering practice.

The central question this lesson addresses is deceptively simple: given a long, straight wire carrying a steady current I, what is the magnitude and direction of the magnetic field B at a perpendicular distance r from the wire? We will derive the answer using both the Biot–Savart law and Ampère's law, compare the two strategies, and build the geometric intuition necessary to tackle more complex configurations.

Before diving into calculations, it is essential to establish the physical principles that underpin both the Biot–Savart law and Ampère's law. The magnetic field produced by a steady current is a vector field that permeates the space around the conductor. Unlike the electric field of a point charge, which radiates outward in all directions, the magnetic field lines form closed loops that encircle the current. Understanding the geometry of these loops and the mathematical tools that describe them is the key to mastering this topic.

The diagram above presents the end-on (cross-sectional) view of a long straight wire. The current I flows out of the page, indicated by the dot symbol (think of the tip of an arrow approaching you). The magnetic field lines form concentric circles centered on the wire, and their direction — counterclockwise here — is determined by the right-hand rule. A critical geometric observation is that B is everywhere tangent to these circles and its magnitude is constant along any single circle of radius r. This azimuthal symmetry is precisely what makes Ampère's law so efficient for this geometry: the integral ∮ B · dl around one of these circles immediately factors into B(2πr).

Derivation via the Biot–Savart Law

The Biot–Savart law expresses the infinitesimal magnetic field dB produced at a field point P by a tiny current element I dl located at a source point on the wire. To obtain the total field from an infinitely long straight wire, one integrates over all current elements from −∞ to +∞ along the wire.

Place the wire along the z-axis and the field point P at perpendicular distance s from the wire in the xy-plane. Parametrise the source position as z' so that dl = dz' ẑ. The displacement vector from the source element to P has magnitude r = √(s² + z'²), and the cross product |dz' ẑ × r̂| = s dz' / $(s² + z'²)^{1/2}$. Substituting into the Biot–Savart expression and integrating z' from −∞ to +∞ yields the standard integral ∫ dz' / $(s² + z'²)^{3/2}$ = 2 / s². After simplification, the result is the well-known formula for the magnitude of B.

Derivation via Ampère's Law

Ampère's law provides the same result with far less algebra when the geometry possesses sufficient symmetry. For an infinitely long straight wire, the magnetic field has only an azimuthal component that is constant on any circle concentric with the wire. Choose an Amperian loop — a circle of radius r in a plane perpendicular to the wire, centered on the wire. Along this loop, B is parallel to dl everywhere and has constant magnitude, so the line integral simplifies immediately.

The most common challenge students encounter with the Biot–Savart derivation is setting up the geometry correctly. The diagram below shows a side view of the infinite wire, a single current element at position z' on the wire, and the field point P at perpendicular distance s. All angles, vectors, and distances are labelled to make the integration transparent.

The crucial trigonometric substitution is z' = s tan φ, which transforms the integral into one over the angle φ from −π/2 to +π/2. After evaluating, the factors of 4π cancel neatly against the prefactor, leaving the familiar result B = μ₀I / (2πs). For a finite wire segment subtending angles θ₁ and θ₂ at the field point, the result generalises to B = (μ₀I / 4πs)(cos θ₁ − cos θ₂), which reduces to the infinite-wire formula when θ₁ → 0 and θ₂ → π.

Students frequently ask which method to use on an exam. The answer depends almost entirely on the symmetry of the current distribution. The table below provides a concise decision guide. In general, attempt Ampère's law first whenever the geometry allows you to identify a loop on which B is constant and parallel (or perpendicular) to dl; resort to Biot–Savart when no such loop exists.

The infinite-straight-wire result is not merely an isolated formula — it is a gateway to several more advanced topics. When Maxwell extended Ampère's law to include the displacement current term (ε₀ ∂E/∂t), the law became valid for time-varying fields as well, ultimately leading to the prediction of electromagnetic waves. In differential form, Ampère–Maxwell's equation reads ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t. The steady-state version (∇ × B = μ₀J) is precisely what we used in integral form in this lesson.

Historically, the force per unit length between two parallel current-carrying wires, F/L = μ₀I₁I₂ / (2πd), served as the basis for the SI definition of the ampere until 2019 when it was redefined in terms of the elementary charge. The straight-wire magnetic field formula thus sits at the intersection of fundamental physics and metrology. As you proceed to study solenoids and Faraday's law, the intuitions you build here — circling field lines, the 1/r dependence, the role of symmetry in simplifying calculations — will transfer directly.