Magnetic Field of Straight Conductor
When electric current flows through a straight wire, it creates a magnetic field that wraps around the wire in concentric circles — much like the rings of a tree trunk when viewed from the end. This is not obvious from everyday experience, because the effect was only discovered in 1820 when Hans Christian Ørsted noticed a compass needle deflecting near a current-carrying wire. Unlike the field of a bar magnet, which runs between poles, the field here has no beginning or end: it forms closed loops centred on the wire. The direction of the circular field can be predicted using the right-hand grip rule: wrap your right hand around the wire with the thumb pointing in the direction of conventional current (positive to negative), and your curled fingers show the direction the field lines circle around the wire. Reversing the current reverses the field direction. This simple rule is enormously useful — it lets you predict whether two parallel wires will attract or repel each other and is the basis of electromagnet design. The strength of the field is not uniform: it decreases as you move further from the wire. Mathematically, the field strength is inversely proportional to the perpendicular distance from the wire. This makes physical sense — the same total field 'threads' through larger and larger circles, so it must spread thinner with distance.
- Why Does Current Create a Field?
- Applying the Biot–Savart Law (and Ampère's Law)
- The Right-Hand Grip Rule
- Effect of Distance and Current
- Units and Scale
Why Does Current Create a Field?
Electric current is moving charge. Moving charges produce magnetic fields — this is a fundamental fact of electromagnetism unified by Maxwell's equations. A steady current in a long straight wire produces a field that, by symmetry, can only depend on the perpendicular distance from the wire and must point in a direction tangential to circles centred on the wire.
Applying the Biot–Savart Law (and Ampère's Law)
The exact formula for the field can be derived using Ampère's Law. Imagine drawing a circular 'Amperian loop' of radius r centred on the wire. The magnetic field B is constant in magnitude around this circle and always tangent to it, so Ampère's Law gives B × (2πr) = μ₀I, which rearranges to B = μ₀I / (2πr). This elegant result shows field strength is proportional to current and inversely proportional to distance.
The Right-Hand Grip Rule
Point your right thumb along the conventional current direction (from + to −). Your fingers curl in the direction of the magnetic field lines. If current flows away from you (into the page, shown as ⊗), the field circles clockwise. If current flows towards you (out of the page, shown as ⊙), the field circles anticlockwise. This is consistent with the cross-product form of the Biot–Savart Law.
Effect of Distance and Current
Doubling the current doubles the field. Doubling the distance from the wire halves the field. This 1/r dependence is weaker than the 1/r² dependence of a point charge's electric field, because the wire is an extended source. In practical terms, this is why electrical cables carrying large currents can interfere with nearby sensitive instruments, even metres away.
Units and Scale
The magnetic field B is measured in tesla (T). A wire carrying 1 A of current produces a field of only 2 × 10⁻⁷ T at a distance of 1 m — far smaller than Earth's field of about 5 × 10⁻⁵ T. To get strong fields you need either very large currents (industrial electromagnets) or very small distances. This is why the effect was not noticed before precise instruments existed.
The Key Equation
- B — Magnetic flux density (Tesla (T)): The strength of the magnetic field at a point; higher values mean a stronger field.
- \mu_0 — Permeability of free space (T·m·A⁻¹): A fundamental constant equal to 4π × 10⁻⁷ T·m·A⁻¹; it describes how strongly a magnetic field is established in a vacuum.
- I — Current (Ampere (A)): The electric current flowing through the wire; larger current produces a stronger field.
- r — Perpendicular distance (Metre (m)): The shortest distance from the wire to the point where the field is measured; field weakens as r increases.
Worked Example
Problem. A long straight wire carries a current of 5.0 A. Calculate the magnetic flux density at a perpendicular distance of 8.0 cm from the wire. (Use μ₀ = 4π × 10⁻⁷ T·m·A⁻¹.)
Answer. The magnetic flux density at 8.0 cm from the wire is 1.25 × 10⁻⁵ T (12.5 μT).
Origins · 1 minHans Christian Ørsted and how this was discovered
In April 1820, Ørsted discovered — reportedly during a lecture demonstration — that a compass needle deflected when placed near a wire carrying electric current. This was the first experimental evidence that electricity and magnetism were connected, sparking the entire field of electromagnetism. He published his findings in Latin in July 1820, prompting Ampère and Biot & Savart to derive the mathematical laws that now govern our understanding of magnetic fields due to currents. Ørsted is also credited with being the first person to isolate aluminium in impure form, in 1825 — two years before the metal was officially 'discovered'. His interests ranged from philosophy and aesthetics to chemistry and physics.
Related Concepts
- Magnetic Field of a Solenoid — A solenoid is formed by winding a straight wire into a coil; its field is a direct extension of the single-wire field analysed here.
- Magnetic Field Lines — Understanding how field lines represent direction and strength is essential for interpreting the circular field pattern around a straight wire.
- Force Between Parallel Conductors — The force between two parallel current-carrying wires arises directly from the circular magnetic field each wire creates around the other.
- Ampère's Law — Ampère's Law is the mathematical tool used to derive the B = μ₀I/(2πr) formula for the straight wire, and applies to all current distributions.