Physics
Magnetic field of currents
Biot–Savart: wire, loop, solenoid, Helmholtz coils.
What happens
Magnetic fields computed straight from the Biot-Savart law: circles hug a straight wire, a loop makes a little dipole, a solenoid traps a nearly uniform field inside itself, and Helmholtz coils are spaced exactly to flatten the field between them. Orbit around to see the 3D line structure of each.
Equations used in the module
Equations dB = (μ₀ I / 4π) · dl × r / r³ (Biot–Savart) B = μ₀ I / (2π r) (wire), B = μ₀ I / (2R) (loop centre) B = μ₀ n I (solenoid), Helmholtz: d = R units: μ₀/4π = 1e-7 T·m/A, lengths in metres; 1 A in a 1 m loop → 0.628 μT at the centre; Earth ≈ 50 μT = 30 A at R = 0.377 m measured (twin): loop sum vs closed form 1e-9; finite ±6 m wire 1.4 % low at r = 1, 5.1 % at r = 2; Helmholtz B(R/4)/B(0) − 1 = −4.20e-3, B(R/2): −5.42e-2; 9-turn solenoid at pitch 0.22: 0.818 of μ₀nI, 1.0014 of the continuous finite winding
