Biot-Savart Law

Core Idea

The Biot-Savart law gives the magnetic field contribution dB from an infinitesimal current element Id l: dB = (μ₀/4π) (Id l × r̂)/r². The total field is obtained by integrating over the entire current distribution. For a long straight wire at distance r, the result is B = μ₀I/(2πr). For a circular current loop of radius R at its center, B = μ₀I/(2R). The permeability of free space μ₀ = 4π × 10⁻⁷ T·m/A.

How It's Best Learned

First verify the formula for a long straight wire by integrating the Biot-Savart law, then use the result without re-deriving it for composite geometries. Always identify the symmetry and dominant field direction before computing the integral.

Common Misconceptions

• Biot-Savart applies to steady currents; it cannot be directly applied to accelerating charges.
• The cross product d l × r̂ determines the direction of dB — do not just look at the magnitude.
• Unlike Coulomb's law, the Biot-Savart integrand is not radially symmetric.

Explainer

The Biot-Savart law is to magnetism what Coulomb's law is to electrostatics: it tells you the magnetic field produced by a source, built up element by element. The key conceptual move is the same one you used in integration — divide the source into infinitesimal pieces, find the contribution from each piece, and sum. Here the "source" is a steady current, and each piece is an infinitesimal current element Idl: a short length of wire dl carrying current I. Each element contributes a tiny field dB, and the total field is the integral of all contributions.

What makes Biot-Savart geometrically richer than Coulomb's law is the cross product dl × r̂. In Coulomb's law, the field from a point charge points radially outward — symmetric in all directions. In Biot-Savart, the field from a current element circles around the wire: it is perpendicular both to the current direction and to the line from the element to the field point. Your prerequisite knowledge of the cross product is essential here — the cross product automatically gives you this perpendicular direction and has magnitude sin θ, so current elements pointing directly toward your field point (θ = 0) contribute nothing. Only elements oriented at an angle to the displacement vector contribute to the field.

The long straight wire result B = μ₀I/(2πr) is the archetype you should derive once and internalize. The strategy: set up coordinates with the wire along the z-axis, write dl = dz ẑ, express r̂ as a function of z and the perpendicular distance r, evaluate the cross product, and integrate from −∞ to +∞. The integral is a standard form. The result says the field circles around the wire at a distance that falls off as 1/r — faster than the 1/r² of Coulomb's law because the field from distant elements cancels transversely. Symmetry is what makes this tractable: you can argue the field must be azimuthal and depend only on distance, then choose a field point and integrate.

For the circular loop at its center, the geometry is simpler because every element dl on the loop is perpendicular to r̂ (which points from the element to the center), so sin θ = 1 for every element. The field from every element points in the same direction (along the axis), and the integral reduces to just integrating dl around the circumference 2πR. This gives B = μ₀I/(2R). When the geometry lacks this perfect alignment, off-axis fields require more work — but the principle is always the same: identify dl, write r̂, take the cross product, integrate.