# Conservative and non-conservative electric field due to to changing magnetic flux

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?

• Break up the very long sentence beginning with “Now” Commented May 17 at 16:28
• please suggest the edit, i will approve it if need Commented May 17 at 16:30
• This is a question about concepts. It's not asking for help with a homework-type task. Commented May 17 at 21:07

• @FUSIONX this can be seen from Ohm's law. If current density $\mathbf j$ is stationary (steady state of current), and there is a rapid jump in conductivity when crossing junctions A,B, there has to be a jump in electric field at those junctions. Mathematically, Ohm's law says $\mathbf j =\sigma \mathbf E$, so a jump in $\sigma$ has to be cancelled by opposite jump in $\mathbf E$, in order for $\mathbf j$ to not have a jump there. A jump in $\mathbf E$ has to be associated with a non-zero charge accumulation, due to the Gauss law for electric field: $\nabla\cdot\mathbf E = \rho/\epsilon_0$. Commented May 17 at 19:49