Biot-Savart Law: Definition, Formula, Derivation & Applications

Biot–Savart law — with formula, worked example and tips for using it in problems you’ll see in GATE, ESE and other engineering exams.

What is the Biot-Savart Law?

The Biot–Savart law gives the magnetic field dB produced at a point in space by a small current element. It is fundamental for calculating magnetic fields produced by arbitrary steady currents.

Formula

The vector form of the law is:

dB = (μ₀ / 4π) * (I · dl × r̂) / r²

Where:

μ₀ = Permeability of free space = 4π × 10^-7 H·m^-1
I = current (A), dl = current element vector
r̂ = unit vector from current element to point, r = distance

Quick Derivation (conceptual)

Consider a small current element dl carrying current I.
The contribution to the field at point P depends on I, the geometry (dl and the angle θ), and the distance r.
Experimental observation and symmetry show the dependence is proportional to I · dl · sinθ / r². Introducing the constant μ₀/4π yields the full expression above.

Common Results (handy formulas)

Infinite straight wire: B = μ₀ I / (2πr)
Center of a circular loop (single turn): B = μ₀ I / (2R)
Solenoid (ideal, length L ≫ R): B = μ₀ n I where n = N/L

Worked Example (solved)

Problem: A circular coil of radius R = 0.1 m has N = 50 turns and carries current I = 5 A. Find the magnetic field at the center.

Solution: For a coil with N turns, B = μ₀ N I / (2R).

μ0 = 4π × 10^-7

B = (4π × 10^-7 × 50 × 5) / (2 × 0.1) = 1.57 × 10^-3 T = 1.57 mT

Biot-Savart vs Ampere

Aspect	Biot-Savart Law	Ampère's Law
Use case	General: any current shape	Best for high-symmetry cases
Complexity	Integral form — often more work	Simpler when symmetry allows
Form	dB ∝ I dl × r̂ / r²	∮B·dl = μ₀ I_enc

Tip: Use Biot–Savart for finite/irregular geometries. Use Ampère's law when the field and path follow a symmetry (infinite wire, solenoid, toroid).

Applications

Design of electromagnets, inductors and transformers
Magnetic field mapping for sensors and MRI coils
Analysis of current-carrying traces in PCB design
Fundamental in antenna theory and electromagnetic compatibility (EMC)