Solenoid Vector Potential

Solenoid Magnetic Vector Potential

Please show that the Magnetic vector potential of a solenoid is consistent with the required values of it’s divergence, curl and laplacian in all regions.

$A_ϕ=μnI/2 s*ϕ ̂ $ inside the and $A_ϕ=μnI/2s R^2*ϕ ̂ $ outside
Cylindrical coordinates $(s,ϕ,z)$
$∇A=\frac{1}{s}*\frac{∂A_s}{∂s}*s ̂+\frac{1}{s}*\frac{∂A_ϕ}{∂ϕ} ϕ ̂+\frac{∂A_z}{∂z}$

$∇×A=\begin{vmatrix}s ̂/s&ϕ ̂&z ̂/s\\∂/∂s&∂/∂ϕ&∂/∂z\\0&sA_ϕ&0)\end{vmatrix}=$
$=-\frac{s ̂}{s}*\frac{(∂(sA_ϕ))}{∂z}+\frac{z ̂}{s} \frac{(∂(sA_ϕ))}{∂s}$

and $∇^2 A=\frac{1}{s}*\frac{∂}{∂s} (s*\frac{∂A}{∂s})$
For solenoid one has
$∇A=0$ ,$∇×A=B$ and $∆A=μ_0*J$

$∇A=\frac{1}{s}*\frac{(∂A_ϕ)}{∂ϕ} ϕ ̂=0$
$∇×A=\frac{z ̂}{s} \frac{(∂(sA_ϕ))}{∂s}=\frac{z ̂}{s}*\frac{∂}{∂s} (μnI/2 s^2 )=μnI=B$
$∇^2 A=\frac{1}{s}*\frac{∂}{∂s} (μnI/2 s)=μnI/2s=μNI/2sL=(μI_{tot})/Area=μJ$

$∇A=\frac{1}{s}*\frac{(∂A_ϕ)}{∂ϕ} ϕ ̂=0$
$∇×A=\frac{z ̂}{s} \frac{(∂(sA_ϕ))}{∂s}=\frac{z ̂}{s}*\frac{∂}{∂s} (μnI/2 R^2 )=0$
$∇^2 A=\frac{1}{s}*\frac{∂}{∂s} (μnI/2 R^2 )=0$ since $J_{outside}=0$