r = xx + zẑ
r′ = r cos ϕ′x + r sin ϕ′ŷ
dl = −rdϕ′ sin ϕ′x + rdϕ′ cos ϕ′ŷ
r − r′ = (x − r′ cos ϕ′)x + r sin ϕ′ŷ + zẑ
dl × (r − r′) = zrdϕ cos ϕ′x + zrdϕ sin ϕ′ŷ + [−xrdϕcosϕ′ + r²dϕ]ẑ

=

with

α = x² + r² + z²
β = 2rx

expand

(α − β cos ϕ′)⁻

rewrite as

α⁻ (1 −x)⁻

use expansion formula (1 − x)⁻ⁿ = 1 + +

1 + cos ϕ′ + cos ²ϕ′

B_r =

B_𝜃 = −