1.4 Ampère's circuital law

Ampère's circuital law

Ampère's circuital law connects magnetic fields to the electric currents that produce them. It plays a role for magnetic fields analogous to what Gauss's law does for electric fields: given enough symmetry, it turns a difficult integral into a simple algebraic equation. The law also forms one of Maxwell's four equations, and its generalized form (with the displacement current) is essential for understanding electromagnetic wave propagation.

Magnetic fields of current distributions

Magnetic field of a straight wire

The magnetic field lines around a long, straight current-carrying wire form concentric circles centered on the wire. You find the direction with the right-hand rule: point your thumb along the current, and your fingers curl in the direction of $\vec{B}$.

The field magnitude falls off as $1/r$, where $r$ is the perpendicular distance from the wire:

$B = \frac{\mu_0 I}{2\pi r}$

This result follows directly from applying Ampère's law with a circular Amperian loop of radius $r$ centered on the wire.

Magnetic field of a circular loop

Field lines from a circular current loop look like a magnetic dipole pattern (similar to a bar magnet). The field is strongest at the center of the loop and along its axis. Use the right-hand rule again: curl your fingers in the direction of the current, and your thumb points along the magnetic moment (the direction of $\vec{B}$ through the center).

At the center of a single loop of radius $R$:

$B = \frac{\mu_0 I}{2R}$

The field is proportional to the current and inversely proportional to the loop radius. Helmholtz coils exploit this geometry by placing two identical loops a specific distance apart to create a region of nearly uniform field between them.

Magnetic field of a solenoid

A solenoid is a tightly wound helical coil. Inside an ideal (infinitely long) solenoid, the magnetic field is remarkably uniform and parallel to the axis. Outside, the field is essentially zero.

Applying Ampère's law to a rectangular loop that straddles the solenoid wall gives:

$B = \mu_0 n I$

where $n$ is the number of turns per unit length and $I$ is the current. Notice that this result is independent of position inside the solenoid and independent of the solenoid's cross-sectional radius. If a core material with permeability $\mu$ fills the solenoid, replace $\mu_0$ with $\mu$.

Mathematical formulation

Integral form

Ampère's law in integral form states:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

• $\vec{B}$ is the magnetic field vector at each point on the closed path (the Amperian loop).
• $d\vec{l}$ is an infinitesimal line element directed along the path.
• $I_{\text{enc}}$ is the total net current threading through any surface bounded by the loop.

The entire strategy for using this law rests on choosing a loop where symmetry makes $\vec{B} \cdot d\vec{l}$ easy to evaluate. On a well-chosen path, $\vec{B}$ is either constant in magnitude and parallel to $d\vec{l}$, or perpendicular to $d\vec{l}$ (contributing zero), or zero altogether.

Enclosed current and sign conventions

The enclosed current is the net current passing through any surface bounded by your chosen loop:

$I_{\text{enc}} = \int_S \vec{J} \cdot d\vec{A}$

where $\vec{J}$ is the volume current density and $d\vec{A}$ is the area element with direction given by the right-hand rule applied to the loop's circulation direction. Currents flowing in the direction of $d\vec{A}$ count as positive; currents flowing opposite count as negative. If multiple wires pass through the loop, you sum their contributions with appropriate signs.

Differential form

Applying Stokes' theorem to the integral form shrinks the loop to an infinitesimal size and yields:

$\nabla \times \vec{B} = \mu_0 \vec{J}$

This says the curl of $\vec{B}$ at any point equals $\mu_0$ times the local current density at that point. Where there is no current, the curl of $\vec{B}$ is zero. This form is particularly useful in problems with continuous current distributions and high symmetry, and it connects directly to the other Maxwell equations in differential form.

Applications of Ampère's law

Step-by-step method for calculating fields

1. Identify the symmetry of the current distribution (cylindrical for a wire, planar for a sheet, solenoidal for a coil, etc.).
2. Choose an Amperian loop that exploits that symmetry. The loop should pass through the point where you want $\vec{B}$, and its shape should make $\vec{B} \cdot d\vec{l}$ constant or zero on each segment.
3. Evaluate the line integral $\oint \vec{B} \cdot d\vec{l}$. On segments where $\vec{B}$ is parallel to $d\vec{l}$ and constant in magnitude, this becomes $B \times (\text{length of that segment})$. On segments where $\vec{B}$ is perpendicular to $d\vec{l}$, the contribution is zero.
4. Determine $I_{\text{enc}}$ by counting (with signs) all currents passing through the surface bounded by the loop.
5. Solve $B \times (\text{effective path length}) = \mu_0 I_{\text{enc}}$ for $B$.

Symmetry considerations

Ampère's law is most powerful when the current distribution has one of these symmetries:

• Cylindrical symmetry (infinite straight wire, coaxial cable): use a circular Amperian loop concentric with the wire.
• Planar symmetry (infinite sheet of current): use a rectangular loop straddling the sheet.
• Solenoidal symmetry (infinite solenoid, toroid): use a rectangular loop for the solenoid or a circular loop threading the toroid.

Without one of these symmetries, Ampère's law still holds as a true statement, but it won't simplify enough to solve for $\vec{B}$. In those cases, you need the Biot-Savart law or numerical methods.

Limitations and assumptions

• Steady currents only. The original form of Ampère's law assumes $\partial \vec{E}/\partial t = 0$. For time-varying fields, you need the displacement current correction (see below).
• Closed loop required. The integration path must form a closed loop. An open path gives no useful result.
• Symmetry required for practical use. The law is always true, but it's only useful for calculating fields when symmetry constrains $\vec{B}$ along the loop.

Displacement current and the Ampère-Maxwell law

The problem with the original law

Consider a charging parallel-plate capacitor. Current flows into one plate and out of the other, but no conduction current crosses the gap between the plates. If you draw an Amperian loop around the wire and choose a flat surface, you get $I_{\text{enc}} = I$. But if you choose a bulging surface that passes between the plates (still bounded by the same loop), you get $I_{\text{enc}} = 0$. Ampère's law gives two different answers for the same loop, which is a contradiction.

Maxwell's fix

Maxwell resolved this by adding a displacement current term. The changing electric flux between the capacitor plates acts as an effective current:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d}{dt} \int_S \vec{E} \cdot d\vec{A}$

The quantity $\varepsilon_0 \frac{d}{dt} \int \vec{E} \cdot d\vec{A}$ is the displacement current $I_d$. For the capacitor example, the growing electric field between the plates produces a displacement current exactly equal to the conduction current in the wire, restoring consistency regardless of which surface you choose.

In differential form, the Ampère-Maxwell law becomes:

$\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$

Connection to electromagnetic waves

The displacement current term is what makes electromagnetic waves possible. Faraday's law says a changing $\vec{B}$ produces $\vec{E}$. The Ampère-Maxwell law says a changing $\vec{E}$ produces $\vec{B}$. Together, these two laws allow self-sustaining oscillations of $\vec{E}$ and $\vec{B}$ that propagate through space at:

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

This prediction, that electromagnetic waves travel at the speed of light, was one of the great triumphs of Maxwell's theory and confirmed that light itself is an electromagnetic wave.

Comparison with the Biot-Savart law

Both Ampère's law and the Biot-Savart law relate magnetic fields to currents, but they're suited to different situations.

|Ampère's law|Biot-Savart law| |---|---|---| |Applicability|High-symmetry current distributions|Any current distribution| | Output | Field along a chosen loop | Field at any specific point in space | | Typical calculation | Algebraic (after exploiting symmetry) | Vector integral over the entire current distribution | |Time-varying fields|Needs displacement current correction|Needs retardation corrections (Jefimenko's equations)| When to use Ampère's law: The problem has obvious symmetry (infinite wire, infinite solenoid, toroid, infinite current sheet), and you want the field magnitude quickly.

When to use Biot-Savart: The geometry lacks symmetry (finite wire, arbitrary loop, off-axis points), or you need the full vector field at a specific location. The trade-off is that Biot-Savart integrals are usually harder to evaluate.

For a finite solenoid, for instance, Ampère's law can't cleanly separate the field contributions, so you'd turn to Biot-Savart (or numerical computation) instead.