Biot–Savart Law (Conceptual)

When electric current flows through a wire, it produces a magnetic field in the space surrounding it. But how strong is that field, and in which direction does it point? The Biot–Savart Law answers precisely these questions by treating any current-carrying conductor as a collection of tiny segments — called infinitesimal current elements — and summing up the magnetic field contribution from each one. Formulated in 1820 by French physicists Jean-Baptiste Biot and Félix Savart, it is the magnetic analogue of Coulomb's Law in electrostatics, giving us a way to calculate the magnetic field produced by any current configuration. The central idea is elegant: each tiny piece of a current-carrying wire, of length d𝓁, contributes a small magnetic field dB at a point in space. This contribution depends on three things — the size of the current, the length of the element, and the distance from the element to the point of interest. Crucially, the direction of dB is perpendicular to both the current direction and the line connecting the element to the field point. This perpendicularity is described mathematically using the cross product, making the Biot–Savart Law inherently three-dimensional. In practice, the Biot–Savart Law is used to derive the magnetic fields of common configurations: a long straight wire, a circular current loop, or a solenoid. Although the full calculation requires integration (summing infinitely many tiny contributions), the conceptual picture is powerful even without the calculus: every current element acts as a tiny magnet-source, and the total field is the vector sum of all such contributions throughout the conductor.

Key takeaways
  • The Current Element
  • The Position Vector r̂
  • The Cross Product and Direction
  • Summing All Contributions
  • The Role of μ₀

The Current Element

Imagine chopping a wire into extremely small pieces. Each piece has a length d𝓁 (an infinitesimally small length) and carries current I. We represent this piece as the vector Id𝓵, pointing in the direction of current flow. Think of it like a tiny arrow sitting on the wire.

The Position Vector r̂

For each current element, draw a line from that element to the field point P — the place where you want to know the magnetic field. The distance along this line is r, and the unit vector pointing from the element towards P is r̂. The magnitude of dB depends on 1/r², just like gravitational and electric fields — the further away P is, the weaker the contribution.

The Cross Product and Direction

The direction of dB is given by the cross product d𝓵 × r̂. This cross product always produces a vector perpendicular to both the current direction and the line from element to P. Use the right-hand rule: point your fingers along the current, curl them towards r̂, and your thumb points in the direction of dB. This is why magnetic field lines form circles around a straight wire — every element contributes a field that loops around the wire.

Summing All Contributions

The total magnetic field B at point P is found by integrating (summing) the contributions dB from every element along the entire wire. For a long straight wire, this integration gives B = μ₀I / (2πd), where d is the perpendicular distance from the wire. For a circular loop of radius R at its centre, it gives B = μ₀I / (2R). Each configuration requires its own integral, but the underlying principle — sum the tiny contributions — is always the same.

The Role of μ₀

The constant μ₀ = 4π × 10⁻⁷ T·m/A is called the permeability of free space. It plays the same role in magnetism that ε₀ (permittivity) plays in electrostatics — it sets the scale of how strong a magnetic field a given current produces in a vacuum. A larger μ₀ would mean currents produce stronger fields; its actual value reflects the fundamental nature of electromagnetism.

The Key Equation

dB=μ04πI dℓ×r^r2d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I\, d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}
  • d\mathbf{B} — Infinitesimal magnetic field (Tesla (T)): The tiny contribution to the magnetic field at point P from one current element.
  • \mu_0 — Permeability of free space (T·m/A): The magnetic constant, equal to 4π × 10⁻⁷ T·m/A, relating magnetic field strength to current in a vacuum.
  • I — Electric current (Ampere (A)): The steady current flowing through the wire segment.
  • d\boldsymbol{\ell} — Infinitesimal length element (Metre (m)): A tiny vector element of the wire, pointing in the direction of current flow.
  • \hat{\mathbf{r}} — Unit position vector (Dimensionless): A unit vector pointing from the current element towards the field point P.
  • r — Distance (Metre (m)): The distance from the current element to the field point P.

Worked Example

Problem. A long, straight wire carries a current of 5 A. Using the result derived from the Biot–Savart Law, calculate the magnitude of the magnetic field at a perpendicular distance of 0.10 m from the wire.

B=μ0I2πd=4π×10−7×52π×0.10=1.0×10−5 TB = \frac{\mu_0 I}{2\pi d} = \frac{4\pi \times 10^{-7} \times 5}{2\pi \times 0.10} = 1.0 \times 10^{-5}\text{ T}

Answer. The magnetic field at 0.10 m from the wire is 1.0 × 10⁻⁵ T (10 μT), directed in concentric circles around the wire according to the right-hand rule.

Figure
Diagram
Wire (current I)I ↑Id𝓵rPB field circlesdB ⊥ to bothId𝓵 and r̂d (perp. distance)Circular B-fieldlines (right-hand rule)
Origins · 1 minJean-Baptiste Biot and how this was discovered
JB
Jean-Baptiste Biot
1774–1862 · French

Together with Félix Savart, Biot conducted the landmark 1820 experiments that established the quantitative relationship between electric current and the magnetic field it produces. Within weeks of Ørsted's discovery that currents deflect compass needles, Biot and Savart derived the law bearing their names, showing that the field from a long wire decreases as 1/r and is directed perpendicularly to the wire. This was one of the first precise mathematical laws of electromagnetism. Biot was also a pioneering aeronaut — he ascended in a hot-air balloon with Gay-Lussac in 1804 to study Earth's atmosphere and magnetic field at high altitude, making him one of the earliest scientist-explorers of the upper atmosphere.

  • Magnetic Field of a Straight Wire — The field of a long straight wire is the most important direct application of integrating the Biot–Savart Law, giving B = μ₀I/(2πd).
  • Ampère's Law — Ampère's Law is a complementary method for finding magnetic fields due to currents, often simpler to apply when there is high symmetry, and is closely related to the Biot–Savart Law.
  • Magnetic Force on a Current-Carrying Conductor — Understanding how a magnetic field is created by a current (Biot–Savart) pairs naturally with understanding the force that an external field exerts on a current-carrying wire.
  • Solenoids and Toroids — The uniform magnetic field inside a solenoid is derived by applying the Biot–Savart Law to each loop, making solenoids a key worked example of the law in action.