Consider this situation in which a uniform cross section conducting neutral ring is placed in a changing magnetic field which is given by $B=kt(-\hat{k})$ ($k$ is a constant and $t$ is time).
The upper half has a resistivity twice that of lower half. Now, according to Faraday's law of electromagnetic induction, the net emf enduced in the ring will be of magnitude $k\pi r^2.$ Due to this emf, a net current will flow through the uniform cross section of the ring which will be equal for lower as well as upper half hence current density $J$ will be equal in both halves. So, (Electric field)/(resistivity)=constant (from the equation $J=\sigma E$), but since the resistivity in lower and upper halfs are different, the electric field must be different, too. But, in such cases, the non-conservative electric field is given by the formula $= -(r/2)\times \mathrm{d}\Phi_b/\mathrm{d}t$ (rate of change of magnetic flux), which is same for both halves. How does the electric field change in both halves? Is there a conservative electric field induced?