Magnetic Field of Infinite Wire

The magnetic field of an infinitely long straight current-carrying wire is one of the most important standard results in magnetostatics. It represents the simplest and most symmetrical application of the Biot-Savart Law, and its result — concentric circular field lines with magnitude inversely proportional to radial distance — directly forms the basis for understanding force between parallel conductors, inductance, and shielding in electromagnetic design.

Core Concept Explanation

When current flows through a long straight conductor, the resulting magnetic field lines form perfect concentric circles centered on the wire. The field direction at any point is tangential (in the φ direction in cylindrical coordinates), and its magnitude depends only on the perpendicular distance r from the wire, not on position along the wire's axis. This cylindrical symmetry makes the infinite wire the simplest geometry to analyze.

Physically, this result arises because contributions from current elements above and below any given cross-section cancel each other's axial and radial components, leaving only the tangential component. The field is strongest close to the wire and weakens with distance — a hallmark of the 1/r dependence that appears in 2D cylindrical geometries (as opposed to the 1/r² of a point source in 3D).

This field pattern is directly observed when iron filings are placed around a current-carrying wire — they align along circular paths centered on the wire. The physical picture is: the wire is a linear source spreading magnetic influence radially in a cylindrical fashion.

Mathematical Expression

Using Ampere's Circuital Law with a circular Amperian loop of radius r centered on the wire:

∮ H · dL = I_enc → H · (2πr) = I → H = I / (2πr)

The result in vector form (cylindrical coordinates) is H = (I / 2πr) · aφ. The direction aφ follows the right-hand rule: curl the fingers of the right hand in the direction of H when the thumb points in the direction of current.

The magnetic flux density is B = μ₀H = μ₀I / (2πr) in free space. The field falls as 1/r, not 1/r² — this is because the source is a line (one-dimensional), not a point (zero-dimensional). Each additional ring of length at distance r spreads over a circumference of 2πr, giving the 1/r decay.

Practical Understanding

The infinite wire result is directly used in computing the force per unit length between two parallel current-carrying wires. If two wires carry currents I₁ and I₂ separated by distance d, the force per unit length is F/L = μ₀ I₁ I₂ / (2πd). This is the principle behind the original SI definition of the ampere and is tested frequently in GATE.

In power systems, magnetic field calculations near transmission lines use this formula with actual finite lengths approximated as infinite when the observation point is much closer to the wire than to either end. In PCB design, magnetic coupling between adjacent traces is estimated using the same principle.

Given:
A long straight wire carries I = 10 A.
Find H at r = 5 cm from the wire.

Why this formula applies:
Infinite wire with full cylindrical symmetry — Ampere's Law gives H = I/(2πr).

Formula:
H = I / (2π × r)

Substitution:
H = 10 / (2π × 0.05)

Calculation:
H = 10 / 0.3142
H = 31.83 A/m

Final Answer:
H ≈ 31.83 A/m  in the φ (tangential) direction around the wire

Exam Tip: H from an infinite wire falls as 1/r (not 1/r²). If two wires carry currents in the same direction, they attract; opposite directions, they repel. The force formula is F/L = μ₀I₁I₂/(2πd) — always derive it from H × I interaction, not by memory alone.

Mechanism — How the 1/r Dependence Arises

• The Amperian loop is a circle of radius r centered on the wire. By symmetry, H is constant on this circle and tangential.
• ∮H · dL = H × 2πr = I_enc = I, directly giving H = I/(2πr).
• As r doubles, the loop circumference 2πr doubles, so H halves — this is the physical meaning of 1/r dependence.
• The field is purely in the φ direction; there is no radial or axial component due to symmetry.
• For a wire of finite length, the angles at the ends must be accounted for using Biot-Savart, and the 1/r result no longer holds exactly.

Quick Revision

• Infinite straight wire: H = I/(2πr) in the φ direction (cylindrical coordinates).
• Field decreases as 1/r — line source geometry (not 1/r² as in point source).
• Direction: right-hand rule — thumb along I, fingers curl in direction of H.
• B = μ₀H = μ₀I/(2πr) in free space.
• Force per unit length between two parallel wires: F/L = μ₀I₁I₂/(2πd).
• Exam trap: This result assumes infinite length. For finite wires, use Biot-Savart with angle corrections.
• Ampere's Law works here because of perfect cylindrical symmetry — H is constant and tangential on the Amperian circle.