993 B
Chapter 5 - Electrodynamics with Moving Charges
Section 5.1 - Currents in Steady-State Regine
We want to work in a steady-state system. Thus, we restrict ourselves to currents that do not change in time.
With math, we see that \div \vb{J}(\vb{r}) = -\frac{\partial \rho(\vb{r})}{\partial t}. Since we are only considering a steady-state system, \div \vb{J}_e = \div \vb{J}_m = 0.
Definition. The conductance of a material is G = \frac{1}{R}, where R is the resistance of a material.
For a wire of uniform cross-sectional area, we see that G = \sigma \frac{A}{L}, where A is the cross-sectional area, L is the length of the wire, and \sigma is the conductivity of a wire. Inverted, we see that R = \rho \frac{L}{A}, where \rho = \frac{1}{\sigma} is the resistivity of the wire.
Definition. Ohm's Law can be written as I = G V, or inverted, V = IR. In a wire, we see that current density \vb{} = \frac{I}{A} = \sigma \frac{V}{L} = \sigma \vb{E}