The Biot-Savart Law

How steady currents generate magnetic fields — the magnetostatic analog of Coulomb's law

ElectromagneticsElectrical Engineering Year 2Free preview

⏱️ About 18 min

Coulomb's law told us how a static charge creates an electric field. But what creates a magnetic field? The answer is moving charge — current — and the law that quantifies it has the same inverse-square skeleton as Coulomb's, with a twist: the field points sideways, not radially.

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The big idea: The Biot-Savart law gives the magnetic field contribution from a current element I dl as dB = (μ₀/4π)(I dl × r̂)/r². Like Coulomb's law, it is inverse-square; unlike Coulomb's law, the field direction is perpendicular to both the current and the line to the field point, determined by the cross product dl × r̂.

🎯 By the end, you'll be able to

State the Biot-Savart law and identify each term, including the permeability of free space μ₀
Compare and contrast the Biot-Savart law with Coulomb's law for electric fields
Apply the right-hand rule to determine the direction of the magnetic field from a current element
Derive the magnetic field at the center of a circular current loop: B = μ₀I/(2R)

📎 Helpful to know first

Current Density, the Continuity Equation & Ohm's Law in Point Form

From Currents to Magnetic Fields

In the previous lesson, we saw that steady currents flowing through conductors are described by the vector field J, which is divergence-free (∇·J = 0) and related to the local electric field by J = σE. We now ask: moving charges constitute a current, and currents are known to produce magnetic fields. How do we compute the magnetic field produced by a given current distribution? The answer is the Biot-Savart law — the magnetostatic counterpart of Coulomb's law.

The Coulomb Analogy

Recall Coulomb's law for a point charge: the electric field from a charge element dQ at distance r is dE = k(dQ/r²)r̂, pointing radially outward from the charge. The Biot-Savart law has the same inverse-square structure, but the source is not a static charge — it is a current element I dl, a small segment of a current-carrying wire. The field contribution dB from this element is proportional to I dl and falls off as 1/r², but its direction is not radial. Instead, it points perpendicular to both dl and the line from the source to the field point, determined by the cross product dl × r̂.

\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I\, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2} \]

The Biot-Savart law: magnetic field contribution from a current element I dl at distance r.

\[ \mu_0 = 4\pi \times 10^{-7} \text{ T·m/A} \]

Permeability of free space (equivalently H/m).

✨ Coulomb vs. Biot-Savart

Compare the two laws side by side. Coulomb: dE ∝ dQ/r², direction radial (along r̂). Biot-Savart: dB ∝ I dl/r², direction perpendicular (along dl × r̂). Both are inverse-square, both involve a fundamental constant (ε₀ vs. μ₀), but the magnetic field has a fundamentally different geometric character — it circles around the current rather than radiating from it.

Direction via the Right-Hand Rule

The cross product dl × r̂ determines the direction of dB. Point the fingers of your right hand in the direction of dl (the current flow), curl them toward r̂ (the vector from the current element to the field point), and your thumb points in the direction of dB. Equivalently, grasp the wire with your right thumb pointing in the direction of current flow, and your fingers curl in the direction of B. This is one of the most important directional rules in all of electromagnetics.

Integrating Over the Current Path

Just as Coulomb's law gives dE from a charge element and we integrate over the full charge distribution to get E, the Biot-Savart law gives dB from a current element and we integrate over the full current path to get B. For simple geometries — straight wires, circular loops, solenoids — symmetry often lets us evaluate the integral analytically.

\[ \mathbf{B} = \frac{\mu_0}{4\pi} \oint \frac{I\, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2} \]

Total magnetic field from a closed current loop, obtained by integrating the Biot-Savart law around the path.

Field at the Center of a Circular Loop

As our first application, consider a circular loop of radius R carrying current I. We want the magnetic field at the center of the loop. Pick any current element I dl on the loop. The vector r̂ from this element to the center points radially inward, perpendicular to dl (which is tangent to the circle). So |dl × r̂| = dl, and the direction of dB is perpendicular to the plane of the loop.

Crucially, every element on the loop contributes a dB in the same direction at the center: all are perpendicular to the plane and parallel to each other. The vector integral therefore collapses to a simple scalar integral of magnitudes.

\[ B = \frac{\mu_0}{4\pi} \int_0^{2\pi} \frac{I\, R\, d\phi}{R^2} = \frac{\mu_0 I}{4\pi R} \int_0^{2\pi} d\phi \]

Setting up the integral at the loop center: dl = R dφ, r = R, and |dl × r̂| = dl.

\[ B = \frac{\mu_0 I}{2R} \]

Magnetic field at the center of a circular current loop of radius R.

🔑 Why the Integral Collapses

The integral evaluates to 2π because every dB points in the same direction at the center. If we were computing B at a point off-center or on the axis, the cross-product directions would not all align, and the integral would be considerably more involved. The center is the simplest case, and it already gives a useful, memorable result.

📝 Worked example: A circular current loop of radius R = 0.05 m carries current I = 2 A. Find the magnetic field magnitude at the center of the loop.

Use the derived result B = μ₀I/(2R).
Substitute: B = (4π×10⁻⁷)(2)/(2×0.05) = (1.2566×10⁻⁶)(2)/(0.1).
Compute: B = 2.5133×10⁻⁶/0.1 = 2.5133×10⁻⁵ T = 25.13 µT.

✓ 25.13 µT (2.5133×10⁻⁵ T)

Putting the Magnitude in Context

A field of 25 µT is modest — Earth's magnetic field at the surface is roughly 25–65 µT, so this small loop produces a field comparable to the Earth's. To produce stronger fields, you typically need many turns (a coil) or much larger currents.

✏️ Practice: A circular current loop of radius R = 0.1 m carries current I = 3 A. Find the magnetic field B at the center of the loop.

µT

Solution

Use B = μ₀I/(2R).
Substitute: B = (4π×10⁻⁷)(3)/(2×0.1) = (1.2566×10⁻⁶)(3)/(0.2).
Compute: B = 3.7699×10⁻⁶/0.2 = 1.885×10⁻⁵ T = 18.85 µT.

⚠️ Superposition Still Applies

The Biot-Savart law, like Coulomb's law, obeys superposition. The total B from multiple current sources is the vector sum of the individual contributions — a strategy used extensively for straight wires, solenoids, and toroids in upcoming lessons.

Check your understanding

1. The Biot-Savart law gives dB proportional to which expression?

dB = (μ₀/4π)(I dl × r̂)/r² — the cross product of the current element with the unit vector to the field point, divided by r².

2. At the center of a circular loop of radius R carrying current I, what is B?

At the center, all dB contributions align by symmetry, giving B = μ₀I/(2R).

✅ Key takeaways

The Biot-Savart law gives the magnetic field from a current element: dB = (μ₀/4π)(I dl × r̂)/r², where μ₀ = 4π×10⁻⁷ T·m/A.
Like Coulomb's law, the Biot-Savart law is inverse-square; unlike Coulomb's law, the field direction is perpendicular to both the current element and the line to the field point.
The direction of dB follows the right-hand rule applied to dl × r̂ — the magnetic field circles around the current.
At the center of a circular current loop of radius R, symmetry collapses the vector integral to B = μ₀I/(2R).

➡️ The Biot-Savart law is powerful but requires a line integral for every new geometry. In the next lesson, we will meet Ampere's circuital law — a complementary tool that, like Gauss's law in electrostatics, exploits symmetry to compute magnetic fields with far less effort.

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