7.1 Biot-Savart law

Magnetic field of current-carrying wire

Any wire carrying a steady current produces a magnetic field in the surrounding space. The strength and direction of that field depend on the magnitude and direction of the current, as well as the geometry of the wire. This relationship is what the Biot-Savart law captures mathematically, and it underpins the design of electromagnetic devices like transformers, motors, and generators.

Biot-Savart law

Mathematical formulation

The Biot-Savart law gives the infinitesimal magnetic field $d\vec{B}$ produced at a field point by a small current element $Id\vec{l}$:

$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}$

where:

• $d\vec{B}$ is the infinitesimal magnetic field contribution
• $\mu_0 = 4\pi \times 10^{-7} \; \text{T·m/A}$ is the permeability of free space
• $I$ is the steady current through the wire
• $d\vec{l}$ is an infinitesimal directed length element along the wire (pointing in the direction of current flow)
• $\hat{r}$ is the unit vector pointing from the current element to the field point
• $r$ is the distance between the current element and the field point

To find the total magnetic field from an entire wire or current distribution, you integrate over all current elements:

$\vec{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\vec{l} \times \hat{r}}{r^2}$

This integral is a vector integral, so you need to keep track of directions carefully at every step.

Infinitesimal current element

The key idea is that you break the wire into tiny segments $d\vec{l}$, each of which acts as a point-like source of magnetic field. Each segment contributes a small $d\vec{B}$, and you sum (integrate) all these contributions along the entire wire to get the total field. This is a direct application of the superposition principle to magnetostatics.

Cross product and the right-hand rule

The cross product $d\vec{l} \times \hat{r}$ controls both the direction and part of the magnitude of $d\vec{B}$.

Direction: Use the right-hand rule. Point your fingers along $d\vec{l}$ (the current direction), then curl them toward $\hat{r}$. Your thumb points in the direction of $d\vec{B}$.

For a long straight wire, this means the field lines wrap around the wire in concentric circles. You can use a simpler version of the right-hand rule here: point your right thumb along the current, and your fingers curl in the direction of $\vec{B}$.

Magnitude: The cross product contributes a factor of $\sin\theta$, where $\theta$ is the angle between $d\vec{l}$ and $\hat{r}$:

$|d\vec{l} \times \hat{r}| = dl \sin\theta$

Notice that when $\theta = 0$ (field point directly ahead of or behind the current element), the contribution vanishes. The contribution is maximum when $\theta = 90°$.

Applications of Biot-Savart law

Magnetic field of a long straight wire

For an infinitely long straight wire carrying current $I$, integrating the Biot-Savart law over the entire wire gives:

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

where $s$ is the perpendicular distance from the wire. The field lines form concentric circles centered on the wire, and the magnitude falls off as $1/s$.

Sketch of the derivation:

1. Place the wire along the $z$-axis. Choose a field point at perpendicular distance $s$.
2. A current element at position $z'$ on the wire gives $r^2 = s^2 + z'^2$ and $\sin\theta = s / \sqrt{s^2 + z'^2}$.
3. Integrate $dB = \frac{\mu_0 I}{4\pi} \frac{s \, dz'}{(s^2 + z'^2)^{3/2}}$ from $z' = -\infty$ to $+\infty$.
4. The standard integral evaluates to $2/s$, yielding the result above.

For a finite wire segment subtending angles $\theta_1$ and $\theta_2$ as measured from the perpendicular, the result generalizes to:

$B = \frac{\mu_0 I}{4\pi s}(\sin\theta_2 - \sin\theta_1)$

This finite-wire formula is one you'll use repeatedly in problem sets.

Magnetic field of a circular loop

On the axis of a circular loop of radius $R$ carrying current $I$, the Biot-Savart law gives:

$B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}$

where $z$ is the distance along the axis from the center of the loop. At the center of the loop ($z = 0$), this simplifies to:

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

The field at the center points along the axis of the loop, perpendicular to the plane of the current. Its direction follows from the right-hand rule: curl your right-hand fingers in the direction of the current, and your thumb points along $\vec{B}$.

Note that the field is not uniform everywhere inside the loop. It's only approximately uniform very close to the center. The on-axis formula shows the field falls off as $1/z^3$ at large distances ($z \gg R$), which is the signature of a magnetic dipole.

Magnetic field of a solenoid

A solenoid is a tightly wound helical coil. While you can derive the solenoid field by superposing contributions from many circular loops using the Biot-Savart law, in practice Ampère's law is far more efficient here. The result for an ideal (infinitely long) solenoid is:

$B = \mu_0 n I$

where $n$ is the number of turns per unit length. The field is uniform and directed along the solenoid axis inside, and essentially zero outside. This is one of the cleanest results in magnetostatics.

Relationship to Coulomb's law

The Biot-Savart law for magnetostatics is structurally analogous to Coulomb's law for electrostatics. Both are inverse-square laws that require integration over a source distribution:

The cross product in the Biot-Savart law is the crucial difference. It means magnetic fields always point perpendicular to the plane defined by the current element and the displacement vector, unlike electric fields which point radially from charges.

Superposition principle

The Biot-Savart law relies on the superposition principle: the total magnetic field at any point is the vector sum of contributions from every current element in the system. This holds because Maxwell's equations (and hence the Biot-Savart law) are linear in the sources.

In practice, this means you can compute the field from each wire or current segment independently and then add the results as vectors. For complicated geometries, this is exactly what the integral does.

Ampère's circuital law vs. Biot-Savart law

Both laws describe how steady currents produce magnetic fields, but they're suited to different situations.

Ampère's law ($\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$) is powerful when the geometry has enough symmetry to pull $\vec{B}$ out of the integral. It works cleanly for infinite straight wires, infinite solenoids, and toroidal coils. Outside these high-symmetry cases, it's not directly useful for computing $\vec{B}$.

Biot-Savart law is fully general. It applies to any steady current distribution regardless of symmetry. It's the tool you reach for when the geometry is irregular: a finite wire segment, an oddly shaped loop, or a current distribution without a convenient Amperian loop.

The Biot-Savart law is more fundamental in the sense that you can derive Ampère's circuital law from it (by taking the curl of $\vec{B}$ and applying vector identities). Ampère's law, in turn, is one of Maxwell's equations and generalizes beyond magnetostatics when the displacement current term is included.

Limitations and assumptions

Steady currents only. The Biot-Savart law assumes $\partial \vec{J}/\partial t = 0$. If currents are time-varying, the fields they produce include radiation effects and displacement current contributions that the Biot-Savart law does not capture. You need the full set of Maxwell's equations (or the retarded potentials, specifically the Jefimenko equations) for time-dependent problems.

Classical and continuous. The law treats current as a continuous, classical quantity. At atomic scales, quantum electrodynamics replaces this classical picture. For most macroscopic and mesoscopic problems in this course, the classical treatment is perfectly adequate.

Equivalent volume current form. For volume current densities $\vec{J}$, the Biot-Savart law generalizes to:

$\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}') \times \hat{\mathscr{r}}}{\mathscr{r}^2} \, d\tau'$

where $\vec{\mathscr{r}} = \vec{r} - \vec{r}'$ is the separation vector and $d\tau'$ is the volume element. This is the form you'll encounter most often in Electromagnetism II, since real current distributions are three-dimensional.