# Inhomogeneous conductor

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

The connection between the electric field and the current density (current per unit area $j= \frac{dI}{dA}$) is given by $\vec{j}(r)=\frac{1}{\rho(r)} \vec{E}(r)$ This relation is equivalent to the traditional Ohm's Law $V=I R$ 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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