Answer:
B = μ₀ I / 2π L
Explanation:
The biot-Savart law has the equation
B = μ₀ I / 4π ∫ ds * r ^ / r²
Where the vector product gives the direction of the field, ds is taken along the wire length
ds * r ^ = ds sin θ
B = μ₀ I / 4π I ds sinθ / r²
r is the distance from current element to the point of interest, to make the calculation easier suppose that the wire is on the x-axis and the calculation point on the Y-axis at a distance L
ds = dx
We have to write the integral in terms of a single integration variable, if we use geometry, we can write everything in function angle (θ)
sin θ = L / r
r = L / sin θ = L csc θ
tan θ = L / x
x = L / tan θ
dx = L csc²θ dθ
substitutions and calculate
B = μ₀ I / 4π I (L csc²θ dθ) sinθ / (L cscθ)²
B = μ₀ I / 4π I dθ sinθ / a
B = μ₀ I / 4π I dθ sinθ / a
B = μ₀ I 4π L (cos T1-cosT2)
If the wire is very long θ1 = 0 t θ2 =π (cos θ1 -cosθ2) = 2
B = μ₀ I / 2π L
This is the solution
