Electrostatics Ohm's Law

docs/physics/electrostatics/5-moving-charges.md

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+# 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 $\vb{} = \frac{I}{A} = \sigma \frac{V}{L} = \sigma \vb{E}$