Biot–Savart Law (Applications)

When electric current flows through a wire, it creates a magnetic field in the surrounding space. The Biot–Savart Law is the fundamental rule that tells us exactly how strong that field is and in which direction it points. Named after French physicists Jean-Baptiste Biot and Félix Savart, the law treats a current-carrying wire as a collection of tiny current elements, each contributing a tiny magnetic field. By summing all those contributions — using calculus — we can find the total field at any point in space. This mirrors how Coulomb's Law works for electric charges, but applies to moving charges (currents) instead. The two most important applications of the Biot–Savart Law at this level are the magnetic field produced by a long straight wire and the magnetic field at the centre of a circular loop of wire. The straight-wire result is used everywhere in electrical engineering — from calculating forces between overhead power cables to understanding how circuit boards generate interference. The circular-loop result is the building block for more complex devices: stack many loops together and you get a solenoid, the heart of loudspeakers, MRI machines, and electric motors. Mastering these two cases unlocks a huge range of real physics. Both results follow directly from integrating the Biot–Savart Law. For a long straight wire the field lines form concentric circles around the wire, and the field strength falls off as 1/r with perpendicular distance. For a circular loop the field at the centre points perpendicularly through the loop's plane, and its strength depends on the radius and the current. Understanding both geometries, and knowing when to use which, is the core skill this topic develops.

Key takeaways
  • The Biot–Savart Law — the fundamental expression
  • Straight infinite wire — integrating along the wire
  • Circular loop — integrating around the loop
  • Comparing the two results

The Biot–Savart Law — the fundamental expression

The Biot–Savart Law states that a tiny element of wire of length d𝐥 carrying current I produces a magnetic field d𝐁 at a position 𝐫 away from it. The magnitude is dB = (μ₀/4π)(I dℓ sin θ / r²), where θ is the angle between the current element and the line joining it to the field point. The direction of d𝐁 is perpendicular to both d𝐥 and 𝐫, given by the right-hand rule. Think of each tiny piece of wire as a miniature field source; the total field is the vector sum of all contributions.

Straight infinite wire — integrating along the wire

For a long straight wire, every element contributes a field in the same direction at a perpendicular distance r from the wire. Integrating the Biot–Savart expression from −∞ to +∞ along the wire gives a beautifully simple result: B = μ₀I / (2πr). The field lines are circles centred on the wire. Use the right-hand rule: wrap your right hand around the wire with the thumb pointing in the current direction — your fingers curl in the direction of B. The 1/r dependence means doubling your distance halves the field.

Circular loop — integrating around the loop

For a circular loop of radius R carrying current I, every element of the loop is at the same distance R from the centre, and each element's contribution to the field at the centre points in the same direction (perpendicular to the plane of the loop). The integration is therefore straightforward: B = μ₀I / (2R). The direction again follows the right-hand rule: curl the fingers of your right hand in the direction of current flow around the loop, and your thumb points in the direction of B at the centre. Notice the field is stronger for a smaller loop — tighter curves produce stronger fields.

Comparing the two results

Both formulas involve μ₀ (the permeability of free space) and I, confirming that stronger currents always produce stronger fields. For the straight wire the field decreases with distance r, whereas for the loop the field at the centre depends on the loop's radius R — a geometric property, not a distance to a separate field point. If you have N turns of wire wound into a coil (each of radius R), the fields add: B = μ₀NI / (2R). This is why increasing the number of turns is such an effective way to build stronger electromagnets.

The Key Equation

B=μ0I2πr(straight wire)B=μ0I2R(centre of circular loop)B = \frac{\mu_0 I}{2\pi r} \quad \text{(straight wire)} \qquad B = \frac{\mu_0 I}{2R} \quad \text{(centre of circular loop)}
  • B — Magnetic flux density (Tesla (T)): The strength and direction of the magnetic field at the point of interest.
  • \mu_0 — Permeability of free space (T m A⁻¹): A fundamental constant equal to 4π × 10⁻⁷ T m A⁻¹, relating magnetic fields to the currents that produce them.
  • I — Current (Ampere (A)): The electric current flowing through the wire or loop.
  • r — Perpendicular distance (Metre (m)): The shortest distance from the straight wire to the point where the field is calculated.
  • R — Radius of loop (Metre (m)): The radius of the circular current loop; the field formula applies at the centre of the loop.

Worked Example

Problem. A circular loop of wire has a radius of 5.0 cm and carries a current of 3.0 A. (a) Calculate the magnetic flux density at the centre of the loop. (b) How does this compare with the field 5.0 cm from a long straight wire carrying the same current?

Bloop≈3.8×10−5 T,Bwire≈1.2×10−5 TB_{\text{loop}} \approx 3.8 \times 10^{-5}\,\text{T}, \quad B_{\text{wire}} \approx 1.2 \times 10^{-5}\,\text{T}

Answer. Field at loop centre ≈ 3.8 × 10⁻⁵ T; field from straight wire at 5.0 cm ≈ 1.2 × 10⁻⁵ T. The loop produces a field π times stronger than the straight wire at the equivalent distance.

Figure
Diagram
Biot–Savart Law: Two Key GeometriesIrB circlesB = μ₀I / 2πrIcentreBRB = μ₀I / 2RLong Straight WireCircular Loop
Origins · 1 minJean-Baptiste Biot and how this was discovered
JB
Jean-Baptiste Biot
1774–1862 · French

Jean-Baptiste Biot, working with Félix Savart in 1820, conducted careful experiments measuring the magnetic force on a magnetic pole near a current-carrying wire. They derived the mathematical law — now called the Biot–Savart Law — that quantifies the magnetic field produced by any element of electric current. Their work was prompted by Hans Christian Ørsted's 1820 discovery that a compass needle deflects near a current-carrying wire, and it established the quantitative foundation of electromagnetism. Biot was also a pioneering aeronaut — in 1804 he ascended to an altitude of about 4,000 metres in a hot-air balloon with chemist Joseph Louis Gay-Lussac to study the atmosphere and Earth's magnetic field at high altitude, making him one of the first scientist-balloonists.

  • The Magnetic Field — Understanding what a magnetic field is and how it is represented with field lines is essential before applying the Biot–Savart Law to calculate field strengths.
  • Ampère's Law — Ampère's Law provides an alternative, often simpler, method for calculating magnetic fields from currents in symmetric situations and is closely related to the Biot–Savart approach.
  • Solenoids and Toroids — A solenoid is built from many circular loops; the Biot–Savart result for a single loop is the direct building block for understanding the uniform field inside a solenoid.
  • Force on a Current-Carrying Conductor — The magnetic fields calculated via the Biot–Savart Law exert forces on other current-carrying conductors, linking field calculation directly to observable mechanical forces.