Final answer:
The magnetic field at the point (0,θ,0) for an infinite filament with equation y = |x|tanθ is given by H = 2 * (1 - tan(θ/2)) / (1 + tan(θ/2)). This expression matches the field of a straight infinitely long current filament.
Step-by-step explanation:
The equation for the infinite filament is y = |x|tanθ, where 0 ≤ θ ≤ π/2. To find the magnetic field at the point (0,θ,0), we can use Ampere's law.
First, let's consider a small segment of the filament with length dl at a distance x from the origin. The current through this segment is given by I = i * dl, where i is the current per unit length.
The magnetic field due to this small segment of the filament is given by dH = (μ₀/(4π)) * (I/(x^2+y^2)) * dl * sinφ, where μ₀ is the magnetic constant, x and y are the coordinates of the segment, and φ is the angle between the segment and the line joining the segment and the point (0,θ,0).
By integrating this expression over the entire filament and using the properties of the tangent function, we can eventually arrive at the expression for the magnetic field at the point (0,θ,0) as H = 2 * (1 - tan(θ/2)) / (1 + tan(θ/2)).
This expression agrees with the field of a straight infinitely long current filament.