The Biot–Savart Law

The Biot–Savart law is simply a recipe for the magnetic field a current makes: chop the wire into tiny pieces, find each piece's little field, and add them all up. It's named after Jean-Baptiste Biot and Félix Savart, the two French physicists who measured the field around a wire in 1820. It's what lets you work out the field of any shape of wire, from a power line to an MRI coil. The formula looks dense, but it's just four pieces: $\frac{\mu_0}{4\pi}$ (a fixed constant that sets the units), $I\,d\mathbf{l}$ (one tiny piece of current), $\times\,\hat{\mathbf{r}}$ (the cross product that makes the field circle the wire instead of pointing at it) and $\frac{1}{r^2}$ (fading with distance squared, just like Coulomb's law). Bear with it; the rest of the lesson is adding up pieces.

A compass beside a wire. So far we've treated magnetic fields as given and asked what they push on. Now for the other half: where do they come from? Lay a compass next to a wire and switch on a current. The needle swings to point around the wire, not toward it. Currents make magnetic fields, and the field of a long straight current circles it.

The law. Chop a wire into tiny pieces $d\mathbf{l}$, each pointing along the current. Each piece contributes

where $\mathbf{r}$ runs from the piece to the point where you want the field. Compare Coulomb's law for a bit of charge, $d\mathbf{E} = \frac{1}{4\pi\varepsilon_0}\frac{dq\,\hat{\mathbf{r}}}{r^2}$: the same inverse square and the same "add up the pieces" as in Fields of Continuous Charge, but the cross product makes the magnetic field point around the current instead of away from it. Right-hand rule: thumb along the current, and your fingers curl the way $\mathbf{B}$ circles. Here $\mu_0 = 4\pi\times10^{-7}$ T·m/A, so $\mu_0/4\pi = 10^{-7}$ exactly.

A long straight wire. Put the wire on the $z$-axis and the field point at distance $r$ on the $x$-axis. A piece at height $z$ has $d\mathbf{l} = dz\,\hat{\mathbf{k}}$ and separation $\mathbf{r} = \langle r, 0, -z\rangle$, of length $\sqrt{r^2 + z^2}$, so

Notice every piece pushes the field the same way, $\hat{\mathbf{j}}$, around the wire, so nothing cancels and we can just add sizes. Adding them,

The antiderivative comes from the trig substitution $z = r\tan\alpha$. For a finite piece of wire, stop the integral at its ends: $B = \frac{\mu_0 I}{4\pi r}\left(\sin\alpha_2 - \sin\alpha_1\right)$, where $\alpha_1$ and $\alpha_2$ are the angles from the perpendicular to each end.

The center of a loop. For a circular loop of radius $R$, every piece is the same distance $R$ from the center and $d\mathbf{l}$ is perpendicular to $\hat{\mathbf{r}}$, so every piece contributes $\frac{\mu_0 I\,dl}{4\pi R^2}$ along the axis. The pieces add up to the circumference: $B = \frac{\mu_0 I(2\pi R)}{4\pi R^2} = \frac{\mu_0 I}{2R}$. An arc of angle $\phi$ gives the fraction $\frac{\phi}{2\pi}$ of that.

Along the axis. At height $z$ above the center, each piece is $\sqrt{R^2 + z^2}$ away and still perpendicular to $\hat{\mathbf{r}}$, so $|d\mathbf{B}| = \frac{\mu_0 I\,dl}{4\pi(R^2 + z^2)}$. By symmetry the sideways parts cancel around the loop, and only the axial part survives, a fraction $\frac{R}{\sqrt{R^2 + z^2}}$:

Compare the ring of charge, $E_z = kQz/(z^2 + R^2)^{3/2}$: the same denominator, but the magnetic field is largest at the center instead of zero there, because every piece's contribution points along the axis. Far away, $B_z \approx \frac{\mu_0(I\pi R^2)}{2\pi z^3} = \frac{\mu_0\mu}{2\pi z^3}$ falls like $1/z^3$: the field of a magnetic dipole with moment $\mu = I\pi R^2$, the same shape as an electric dipole's. A bar magnet, a compass needle and the Earth all look like current loops from far away.

Wires push on wires. Wire 1 makes a field $\frac{\mu_0 I_1}{2\pi d}$ at wire 2, which feels $I_2LB$ (Forces and Torques on Currents):

Parallel currents attract and opposite currents repel, which is the reverse of electric charges.

A common mistake. Since every piece's field has size $\frac{\mu_0 I\,dl}{4\pi r^2}\sin\theta$, it's tempting to just add up those sizes. That only works when every piece's field points the same way, as for the straight wire and the loop's center. Off to the side or along a loop's axis, the pieces point different ways, and you have to add them as vectors. On the axis, only the axial parts survive. One smaller slip: mixing up $2\pi r$ (straight wire) with $2R$ (loop center).

When the symmetry is good enough, there's a shortcut. In the next lesson, Ampère's Law gets these fields with no integral at all.

Engineers use the Biot–Savart law wherever they need the field of a particular shape of wire: the coils that steer the beam in an electron microscope, the gradient coils that let an MRI scanner locate signals, the windings of motors and transformers. Power companies check field levels near transmission lines with $\mu_0 I/(2\pi r)$, and the force between parallel conductors sets how strongly switchgear and bus bars must be braced against short-circuit currents.