Electric charges come in two flavors: discrete and continuous. Discrete charges are individual points, like electrons. Continuous charges spread over lines, surfaces, or volumes, like a charged wire or sphere. Understanding both types is key to grasping electric fields.

Calculating electric fields from continuous charge distributions involves dividing them into tiny elements. We then use Coulomb's law to find each element's contribution and integrate over the whole distribution. This method works for line, surface, and volume charges.

Continuous Charge Distributions and Their Electric Fields

Discrete vs continuous charges

• Discrete charges consist of individual point charges (electrons, protons, ions)
  • Each charge is a multiple of the elementary charge $e = 1.602 \times 10^{-19}$ C
• Continuous charge distributions have charges distributed continuously over a line, surface, or volume (charged wire, sheet, sphere)
  • Charge quantization still applies, but the distribution can be approximated as continuous for mathematical convenience

Types of continuous charge distributions

• Line charges distribute charge along a line or curve (one-dimensional)
  • Linear charge density $\lambda$ is the charge per unit length $\lambda = dQ/dl$
  • Charged wire is an example of a line charge
• Surface charges distribute charge over a surface or area (two-dimensional)
  • Surface charge density $\sigma$ is the charge per unit area $\sigma = dQ/dA$
  • Charged sheet or plate is an example of a surface charge
• Volume charges distribute charge throughout a volume (three-dimensional)
  • Volume charge density $\rho$ is the charge per unit volume $\rho = dQ/dV$
  • Charged sphere or cube is an example of a volume charge

Electric fields from charge distributions

• General approach divides the continuous charge distribution into infinitesimal elements
  1. Calculate the electric field contribution from each element using Coulomb's law
  2. Integrate the contributions over the entire charge distribution
• Line charge electric field at point $P$ due to a line charge with linear charge density $\lambda$:
  • $d\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{\lambda dl}{r^2} \hat{r}$
  • Integrate along the line: $\vec{E} = \int d\vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{\lambda dl}{r^2} \hat{r}$
  • $\varepsilon_0$ is the permittivity of free space
• Surface charge electric field at point $P$ due to a surface charge with surface charge density $\sigma$:
  • $d\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{\sigma dA}{r^2} \hat{r}$
  • Integrate over the surface: $\vec{E} = \iint d\vec{E} = \frac{1}{4\pi\varepsilon_0} \iint \frac{\sigma dA}{r^2} \hat{r}$
• Volume charge electric field at point $P$ due to a volume charge with volume charge density $\rho$:
  • $d\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{\rho dV}{r^2} \hat{r}$
  • Integrate over the volume: $\vec{E} = \iiint d\vec{E} = \frac{1}{4\pi\varepsilon_0} \iiint \frac{\rho dV}{r^2} \hat{r}$

Electrostatics and related concepts

• Electrostatics deals with stationary electric charges and their fields
• Electric dipole: a pair of equal and opposite charges separated by a small distance
• Electric flux: a measure of the electric field passing through a given surface
• Divergence: a mathematical operation that describes the outward flux of a vector field from a given point