One of the coolest, yet strangest, features of Gauss's Law is that the electric flux through the imagined Gaussian surface is related to the total amount of charge inside the surface (the “enclosed charge”). For a point charge or multiple point charges, it is fairly easy to find the total amount of enclosed charge - you simply sum the amount of charge that inside the surface you pick. However, if you have a line, sheet, or volume of charge, we need to rely on charge density to find the enclosed charge. These notes will review charge density and show how we find the enclosed charge for various shapes or distributions of charge.

Before we talked about charge density in terms of finding a dQ, where we assumed a uniform (or constant) charge density. We will make that same assumption again here (the more complicated non-uniform charge densities may be covered in upper division courses). For a uniform charge distribution, the charge density tells you how much charge there is per unit length (for 1D lines of charge), per area (for 2D sheets of charge), or per volume (for 3D shapes of charge). For 1D charge distributions, we use $\lambda$ as the charge density (which has units of $C/m$); for 2D charge distributions, we use $\sigma$ as the charge distribution (which has units of $C/m^2$); and for 3D charge distributions, we use $\rho$ as the charge density (which has units of $C/m^3$).

If we know the total charge and the total length/area/volume, we can calculate the charge density using: $$\lambda=\frac{Q_{tot}}{L_{tot}}\:\:\:\:\:\:\:\:\:\:\:\: \sigma=\frac{Q_{tot}}{A_{tot}}\:\:\:\:\:\:\:\:\:\:\:\: \rho=\frac{Q_{tot}}{V_{tot}}$$

The charge density then can be used to figure out how much charge is in a certain amount of length, area, or volume, depending on what kind of charge you have: $$Q=\lambda L=\frac{Q_{tot}}{L_{tot}}L\:\:\:\:\:\:\:\:\:\:\:\: Q=\sigma A=\frac{Q_{tot}}{A_{tot}}A\:\:\:\:\:\:\:\:\:\:\:\: Q=\rho V=\frac{Q_{tot}}{V_{tot}} V$$

As an example, say you have charged a 20 cm piece of tape so it has a charge density of $\lambda = 4.9*10^{-7} C/m$. To find the amount of charge in the bottom 5 cm of tape, you would take: $$ Q = \lambda L$$ $$ Q = 4.9*10^{-7} C/m * 0.05 m$$ $$ Q = 2.45*10^{-8} C $$

Or similarly, to find the total amount of charge in the piece of tape: $$ Q_{tot} = \lambda L_{tot}$$ $$ Q_{tot} = 4.9*10^{-7} C/m * 0.2 m$$ $$ Q_{tot} = 9.8*10^{-8} C $$

Dipole and flux

• Goes back to adding up little bits of charge over space
• Use charge density ideas
• Pick limits so only adding charge inside
• Uniform charge density
• Patterns of fields