H of infinite filament by Biot-Savart

Biot-Savart law: the magnetic field intensity at a point due to a current element I dl is

dH = I dl × a_R / (4π R²) = I dl × R / (4π R³)

where R is the vector from the element to the field point.

Set-up

Let the infinite filament lie along the z-axis carrying current I in the +a_z direction. Find H at P(ρ, φ, 0) (by symmetry, any z).

        z
        ^
        |  element I dz at (0,0,z)
        *
        |\
        | \ R
        |  \
   -----+---*--> ρ   (P at distance ρ)

• Element position: (0, 0, z); field point: (ρ, φ, 0).
• R = ρ a_ρ − z a_z, so R = √(ρ² + z²).
• dl = dz a_z.

Derivation

dl × R = dz a_z × (ρ a_ρ − z a_z) = ρ dz a_φ
dH = I ρ dz a_φ / (4π (ρ² + z²)^(3/2))

Integrate from z = −∞ to ∞:

H = (I ρ / 4π) a_φ ∫₋∞^∞ dz / (ρ² + z²)^(3/2)

Use the substitution z = ρ tan α, dz = ρ sec²α dα, (ρ² + z²)^(3/2) = ρ³ sec³α:

∫ dz/(ρ²+z²)^(3/2) = (1/ρ²) ∫₋π/2^π/2 cos α dα = (1/ρ²)[sin α]₋π/2^π/2 = 2/ρ²

Hence

H = (I ρ / 4π) × (2/ρ²) a_φ

H = (I / 2πρ) a_φ A/m

The field is circular around the wire (right-hand rule), and its magnitude falls as 1/ρ. Check with Ampere's law: ∮H·dl = H(2πρ) = I.