Charge Density

Definition: Charge Density

The charge density $\rho$ at a point $(x,y,z)$ is the infinitely small amount $\mathrm{d}q$ of electric charge contained within an infinitely small volume $\mathrm{d}V$ centered around the point $(x,y,z)$.

$$\frac{\mathrm{d}q}{\mathrm{d}V}$$

NOTATION

$$\rho (x,y,z)$$

Theorem: Electric Charge from Charge Density

The total electric charge in a volume $\mathcal{V}$ is the integral of the charge density over $\mathcal{V}$:

$$Q_{\text{total}}={\int }_{\mathcal{V}}\rho \mathrm{dV}$$

PROOF

TODO