When should I use Ampère's law instead of Biot-Savart?

Ampère's law gives the field with little work when the problem has enough symmetry — such as an infinite wire, solenoid, or toroid — so that the field is uniform along a chosen loop; otherwise the Biot-Savart integral is needed.

Does Ampère's law in its original form always hold?

Only for steady currents; with time-varying fields it must be supplemented by Maxwell's displacement-current term to remain consistent with charge conservation.

Biot-Savart and Ampère's Law

The Biot-Savart and Ampère laws are the two equivalent ways to compute the steady magnetic field of a current distribution.

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Definition

The Biot-Savart law expresses the magnetic field as a superposition of contributions from current elements; Ampère's law states that the circulation of the magnetic field around a closed loop equals the permeability times the enclosed steady current, and the two are equivalent for time-independent currents.

Scope

This topic covers the Biot-Savart law for the field of current elements, Ampère's circuital law in integral and differential form, and their use to find the fields of wires, loops, solenoids, and toroids. It includes the consistency of the two laws for steady currents and the role of symmetry in choosing the simpler approach.

Core questions

How is the field of an arbitrary steady current found by integration?
When does Ampère's law give the field directly from symmetry?
Why must Ampère's law be amended once currents vary in time?

Key concepts

current element
magnetic field of a wire
solenoid
toroid
Amperian loop
curl of the magnetic field
permeability of free space

Key theories

Biot-Savart law: A current element produces a magnetic field perpendicular to the current and the displacement to the field point, decreasing as the inverse square of distance; the total field is the integral over the circuit.
Ampère's circuital law: For steady currents, the closed-loop integral of the magnetic field equals the enclosed current times the permeability, giving the field directly when the geometry is symmetric enough.

Clinical relevance

These laws are used to design solenoids and gradient coils for magnetic resonance imaging, electromagnets, current-carrying coils in motors, and magnetic field calculations in electrical engineering.

History

Following Ørsted's 1820 demonstration, Biot and Savart deduced the inverse-square field law from pendulum measurements of a magnetized needle near a current. Ampère, in the same period, derived the circuital relation and the force between currents, work later generalized by Maxwell.

Key figures

Jean-Baptiste Biot
Félix Savart
André-Marie Ampère

Seminal works

jackson1998
griffiths2017

Frequently asked questions

When should I use Ampère's law instead of Biot-Savart?: Ampère's law gives the field with little work when the problem has enough symmetry — such as an infinite wire, solenoid, or toroid — so that the field is uniform along a chosen loop; otherwise the Biot-Savart integral is needed.
Does Ampère's law in its original form always hold?: Only for steady currents; with time-varying fields it must be supplemented by Maxwell's displacement-current term to remain consistent with charge conservation.