acquiring conductivity through curl of magnetic field density and Ampere's circuital law
Right hand side term of yours \$2\pi\,p\,H=\pi p^2 \sigma E\$ is wrong because it implies conductivity to be constant over the whole section up to p radius.
But this is not true.
It could be turned into an ODE like \$2\pi\,p\,H=\int_0^p 2\pi r\,\sigma(r) E \,\mathrm{d}r\$ to be solved  for \$\sigma(r)\$ though