Inhomogeneous conductor

The resistivity of certain conductor with circular cross-sectional area A=1cm2A= 1 cm^{2} and length L=1mL=1m depends only on the the distance r from the axis of the conductor as ρ(r)=kr2 \rho(r)= \frac{k}{r^{2}} with k=1.2×103Ωm3k=1.2\times 10^{-3} \Omega\cdot m^{3}. What is the equivalent resistance in Ohms of the conductor?

The connection between the electric field and the current density (current per unit area j=dIdAj= \frac{dI}{dA}) is given by j(r)=1ρ(r)E(r) \vec{j}(r)=\frac{1}{\rho(r)} \vec{E}(r) This relation is equivalent to the traditional Ohm's Law V=IR V=I R and is known as the microscopic form of Ohm's Law. Therefore a non-uniform resistivity in a material implies that the current is not uniformly distributed over its cross section.

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