Biot-Savart law

Currents create magnetic fields

Steady electric currents produce magnetic fields. Magnetostatics studies magnetic fields produced by currents that do not change with time. The Biot-Savart law is the magnetic counterpart to Coulomb's law in the sense that it builds a field from source elements.

For a small current element I dec{ell}, the magnetic field contribution at a point is

dec{B}=rac{mu_0}{4pi}rac{I dec{ell} imeshat{r}}{r^2}

Here $mu_0$ is the permeability of free space, $r$ is the distance from the current element to the field point, and $hat{r}$ points from the element toward the field point.

Direction from cross product

The direction of dec{B} is given by the right-hand rule for dec{ell} imeshat{r}. Point your fingers in the direction of current element dec{ell}, curl toward $hat{r}$, and your thumb gives the direction of dec{B}.

The magnetic field circles around current-carrying wires. This circular structure is one major difference between magnetic fields and electrostatic fields from stationary charges.

Field of a long straight wire

For a long straight wire carrying current $I$, the magnetic field at distance $r$ is

B=rac{mu_0 I}{2pi r}

The direction is tangent to circles centered on the wire. Use the right-hand grip rule: point your thumb in the direction of conventional current, and your curled fingers show the magnetic field direction.

This result can be derived by integrating the Biot-Savart law or more simply using Ampère's law.

Field at the center of a circular loop

For a circular loop of radius $R$ carrying current $I$, the magnetic field at the center is

B=rac{mu_0 I}{2R}

For $N$ identical turns,

B=rac{mu_0 NI}{2R}

The direction is along the loop's axis, determined by the right-hand rule. Curl your fingers in the direction of current, and your thumb points in the magnetic field direction through the loop.

Magnetic dipoles

A current loop produces a magnetic field similar in shape to that of a bar magnet. Its magnetic dipole moment is

ec{mu}=IAhat{n}

A loop with more turns has

ec{mu}=NIAhat{n}

This connection between current loops and magnetic dipoles helps explain magnetism in materials.

Field on the axis of a loop

On the axis of a circular current loop, at distance $x$ from the center, the magnetic field is

B(x)=rac{mu_0IR^2}{2(R^2+x^2)^{3/2}}

At $x=0$, this reduces to $mu_0I/(2R)$. Far away, the field resembles that of a magnetic dipole.

Superposition

Magnetic fields obey superposition. The total field from multiple current elements or wires is

ec{B}_{net}=sum_i ec{B}_i

or, for continuous currents,

ec{B}=int dec{B}

As with electric fields, symmetry is often the key to simplifying the calculation.

The big idea

The Biot-Savart law calculates magnetic fields from steady currents by adding contributions from current elements. Its cross product gives magnetic fields their circulating geometry around currents. It provides direct formulas for wires, loops, and magnetic dipoles and prepares the way for Ampère's law.