PX157 - C4d - magnetic field on a long cylindrical wire with a hole

system is not symmetric, but it can be treated as the superposition of two symmetric cylinders:
- cylinder of radius, $a$ , with uniform $J$ centred at $R = 0$
- cylinder of radius, $b$ , with uniform $- J$ centred at $R = R_{0}, ϕ = ϕ_{0}$
using cosine rule, the distance from a point, $p$ , to the centre of the second cylinder:

\begin{aligned} R_{2} & = \sqrt{R^{2} + R_{0}^{2} - 2 R R_{0} \cos (ϕ - ϕ_{0})} \\ {\hat{ϕ}}_{2} & = \frac{R}{R_{2}} \hat{ϕ} - \frac{R_{0}}{R_{2}} (\sin (ϕ - ϕ_{0}) \hat{R} + \cos (ϕ - ϕ_{0}) \hat{ϕ}) \end{aligned}

region 1:

\vec{B} = \frac{μ_{0}}{2 π} [I \frac{R}{a^{2}} \hat{ϕ} - I \frac{R_{2}}{b^{2}} {\hat{ϕ}}_{2}]

region 2:

\vec{B} = \frac{μ_{0}}{2 π} [I \frac{R}{a^{2}} \hat{ϕ} - \frac{I}{R_{2}} {\hat{ϕ}}_{2}]

region 3:

\vec{B} = \frac{μ_{0}}{2 π} [\frac{I}{R} \hat{ϕ} - \frac{I}{R_{2}} {\hat{ϕ}}_{2}]

in typed notes: