Varying current density across non uniform conductor

Question

Suppose we have conductor having non uniform cross sections so j =(sigma)E according to ohms law And we know that current through every cross section is same irrespective of the area while current density changes Now if we extend the formulae it becomes $\frac{I} {A} = \frac{(ne^2t) E}{2m}$ .... Now n e t and m are same throughout the conductor (non uniform cross sections) while for each cross section A(area of cross section) is different so only thing that can change in order to Keep current same thru every cross section of different area is E(electric field) so

1.)Why is electric field changing and how?? 2.) If electric field is diff across each cross section then $dV= -\int{E. dl} $ so potential diff is also different across each cross section of same elemental length (dl)

Please explain me with help of formulas and maths...I get a better glimpse of concepts that way

Answer 1

Your (first) formula is incorrect because $\mathbf{E}$ is not constant within a cross-section. You have $\mathbf{j}=\sigma\mathbf{E}=-\sigma\mathbf{\nabla}U$ (I assume that the conductivity is the same everywhere within the conductor, and $U$ is the potential) and the current continuity condition $\mathbf{\nabla}\mathbf{j}=0$, so we have the Laplace equation $\Delta U=0$ for the potential. One needs to define the boundary conditions and solve the Laplace equation.