Ampère’s Circuital Law

Ampère's Circuital Law is one of the fundamental laws of electromagnetism. It states that the line integral of the magnetic field B around any closed loop is proportional to the total electric current passing through the surface bounded by that loop. In other words, currents are the sources of magnetic fields, and if you trace a closed path around a wire, the cumulative 'push' of the magnetic field around that path is directly linked to how much current threads through it. This makes it the magnetic analogue of Gauss's Law for electric fields. The law is extraordinarily useful because it allows us to calculate magnetic fields in highly symmetric situations — such as around a long straight wire, inside a solenoid, or within a toroid — without doing complicated integration over all space. Instead, we cleverly choose a closed path (called an Amperian loop) that matches the symmetry of the problem, reducing the mathematics to simple algebra. Think of it like choosing the right route on a map: a well-chosen path makes the journey trivial. Ampère's Circuital Law applies strictly to steady (time-independent) currents. James Clerk Maxwell later extended it by adding the displacement current term, making it valid for changing fields too. In its original form, it is a cornerstone of magnetostatics and connects neatly with Faraday's Law and the rest of Maxwell's equations.

Key takeaways
  • The Magnetic Field Forms Closed Loops
  • The Line Integral and the Amperian Loop
  • Enclosed Current: I_enc
  • Choosing the Right Amperian Loop
  • Result for a Long Straight Wire

The Magnetic Field Forms Closed Loops

Unlike electric field lines that start and end on charges, magnetic field lines always form closed loops. Around a straight current-carrying wire, these loops are concentric circles centred on the wire. The direction follows the right-hand rule: wrap your right hand around the wire with the thumb pointing in the current direction — your fingers curl in the direction of B.

The Line Integral and the Amperian Loop

A line integral ∮ B · dl means: travel around a closed path, and at every tiny element dl of that path, take the component of B along the path, multiply by the length of the element, and add them all up. If B is always parallel to dl and constant in magnitude around the loop (as it is for a circular path around a straight wire), this simplifies to B × (circumference) = B × 2πr.

Enclosed Current: I_enc

The right-hand side of Ampère's Law involves only the current that actually passes through the surface bounded by your chosen loop — not any current flowing beside it. If multiple wires thread the loop, add their currents algebraically (currents in the opposite direction subtract). Currents outside the loop contribute nothing to the integral, even though they do affect B locally.

Choosing the Right Amperian Loop

The secret to using this law efficiently lies in symmetry. For an infinite straight wire, choose a circle of radius r centred on the wire: B is tangential and constant everywhere on that circle, so the integral collapses to B(2πr). For a solenoid, choose a rectangular loop that runs along the interior axis and returns outside: B is zero outside and uniform inside, giving B × L on one side. The law does the heavy lifting once the loop is chosen wisely.

Result for a Long Straight Wire

Applying Ampère's Law to a circular Amperian loop of radius r around a long straight wire carrying current I: ∮ B · dl = B(2πr) = μ₀I, so B = μ₀I / (2πr). This result shows that the magnetic field decreases as 1/r with distance from the wire — doubling the distance halves the field. The constant μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space.

The Key Equation

∮CB⋅dl=μ0Ienc\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}
  • \oint_C \mathbf{B} \cdot d\mathbf{l} — Magnetic circulation (T·m (tesla-metres)): The closed line integral of the magnetic field B around the Amperian loop C — effectively the net 'winding' of B along the chosen path.
  • \mu_0 — Permeability of free space (T·m/A): A fundamental constant equal to 4π × 10⁻⁷ T·m/A, describing how strongly a magnetic field is produced by a current in a vacuum.
  • I_{\text{enc}} — Enclosed current (A (amperes)): The net steady current passing through the surface bounded by the Amperian loop, counted positive or negative depending on direction via the right-hand rule.

Worked Example

Problem. A long straight wire carries a steady current of 5.0 A. Calculate the magnetic field at a perpendicular distance of 4.0 cm from the wire.

B=μ0I2πr=4π×10−7×5.02π×0.040=2.5×10−5 TB = \frac{\mu_0 I}{2\pi r} = \frac{4\pi \times 10^{-7} \times 5.0}{2\pi \times 0.040} = 2.5 \times 10^{-5} \text{ T}

Answer. The magnetic field at 4.0 cm from the wire is 2.5 × 10⁻⁵ T (25 μT), directed in concentric circles around the wire according to the right-hand rule.

Figure
Diagram
Ampère's Law: Circular Amperian Loop Around a WireI (out of page)Amperian loop CrB field(tangential)∮ B · dl = μ₀ I_encB × 2πr = μ₀ IRight-hand rule:Thumb → I directionFingers curl → B
Origins · 1 minAndré-Marie Ampère and how this was discovered
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André-Marie Ampère
1775–1836 · French

Ampère was the founder of classical electromagnetism. Within weeks of hearing of Ørsted's 1820 discovery that electric currents deflect compass needles, he conducted a brilliant series of experiments and developed a complete mathematical theory of the relationship between electricity and magnetism. His formulation of what we now call Ampère's Circuital Law showed quantitatively how currents produce magnetic fields, and he introduced the concept of the 'solenoid'. Maxwell later called him the 'Newton of electricity'. Ampère was largely self-taught and had mastered advanced mathematics by age 12 by reading encyclopaedias. He suffered the trauma of his father's execution during the French Revolution, yet went on to transform physics. The SI unit of electric current — the ampere — is named in his honour.

  • Magnetic Field Due to a Straight Wire — Ampère's Law is the most elegant tool for deriving the magnetic field around a long straight wire; understanding the result B = μ₀I/(2πr) is the direct application of the law.
  • Solenoids and Toroids — Ampère's Circuital Law is used to show that the field inside an ideal solenoid is uniform (B = μ₀nI) and to derive the field inside a toroid, making these its most important practical applications.
  • Magnetic Force on a Current-Carrying Conductor — Understanding that currents create magnetic fields (via Ampère's Law) is essential before studying how those fields then exert forces on other current-carrying conductors.
  • Faraday's Law of Electromagnetic Induction — Ampère's Law and Faraday's Law are companion equations in Maxwell's framework — together they describe how changing magnetic and electric fields sustain each other, leading to electromagnetic waves.