Playground
Current-carrying wire with concentric magnetic field rings. Adjust current to see B-field strength change.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Magnetic fieldoutput Magnetic field magnitude at the field point | T | M·T⁻²·I⁻¹ | 0 – 0.1 | |
| Current Current in the wire | A | I | 0.01 – 100 | |
| Distance Distance from the current element to the field point | m | L | 0.001 – 1 |
Deep dive
Derivation
Empirically established by Biot and Savart from force measurements. Consistent with Ampère's law via the Stokes theorem. Can also be derived from the vector potential: B = ∇ × A where A = (μ₀/4π)∫(J/r)dV.
Experimental verification
Original measurements by Biot and Savart on long straight wires (1820). Modern verification via SQUID magnetometers measuring fields from precisely controlled current geometries to parts-per-billion accuracy.
Common misconceptions
- The law gives the field from a single current element — the total field requires integration over the entire circuit
- An isolated current element cannot exist (current must flow in a loop), so the law is always used in integral form
- The direction is given by the right-hand rule — curl fingers from dl toward r̂, thumb points along dB
Real-world applications
- MRI magnet design — Helmholtz and solenoid coil field calculations
- Magnetic field modeling for particle accelerator beam lines
- Induction cooktop heating element design
- Geomagnetic field modeling from ocean-floor current loops
Worked examples
Field at the center of a circular loop
Given:
- I:
- 2
- R:
- 0.05
Find: B at center
Solution
B = μ₀I/(2R) = 1.257×10⁻⁶ × 2 / (2 × 0.05) = 2.513×10⁻⁵ T = 25.13 μT
Field of a long straight wire
Given:
- I:
- 10
- r:
- 0.05
Find: B at 5 cm from the wire
Solution
B = μ₀I/(2πr) = 1.257×10⁻⁶ × 10 / (2π × 0.05) = 4.0×10⁻⁵ T = 40 μT
Scenarios
What if…
- scenario:
- What if the loop has N = 100 turns?
- answer:
- Field multiplies by N: B = 100 × 25.13 μT = 2.513 mT. Coils stack fields linearly — the principle behind electromagnets.
- scenario:
- What if the current reverses direction?
- answer:
- Field direction reverses but magnitude stays the same. The right-hand rule now gives the opposite direction.
- scenario:
- What if you move to a point on the axis but away from center?
- answer:
- B = μ₀IR²/[2(R² + z²)^(3/2)]. At z = R, B drops to ~35% of center value. At z >> R, it falls as 1/z³ (dipole field).
Limiting cases
- condition:
- r → ∞
- result:
- B → 0
- explanation:
- Magnetic field of a finite current loop falls off as 1/r³ (dipole) at large distances.
- condition:
- I → 0
- result:
- B → 0
- explanation:
- No current means no magnetic field.
- condition:
- R → 0 (loop radius)
- result:
- B → ∞ at center
- explanation:
- Shrinking the loop while keeping current fixed concentrates the field — in reality, wire thickness limits this.
Context
Jean-Baptiste Biot & Félix Savart · 1820
Weeks after Ørsted showed a current deflects a compass, Biot and Savart measured the force and deduced the inverse-square dependence on distance — the magnetic analog of Coulomb's law.
Hook
How does a coil of wire become a magnet when you run current through it?
Find the magnetic field at the center of a circular loop of radius 0.05 m carrying 2 A of current.
Dimensions: [B] = [μ₀]·[I]·[r]⁻¹ → (T·m/A)(A)(m⁻¹) = T ✓
Validity: Valid for steady (DC) currents in vacuum. For time-varying currents, use the Jefimenko equations or full Maxwell's equations. In magnetic media, μ₀ is replaced by μ₀μᵣ.