# Inhomogeneous conductor

**Electricity and Magnetism**Level 4

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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