8.3 Electric Fields

Assume there is a source charge ${Q}$ and a point charge ${q}$. The electric force on the point charge q can be derived by Coulomb's law and is proportional to q. The ratio of the force to the ${q}$ is independent of the ${q}$ and has the same value no matter what the point charge is. This ratio of ${\vec{F}/q}$ is known as the electric field ${E}$. The electric field is a vector quantity that points in the same direction as the force a positive charge would feel at that location.

The force on a point ${q}$ in an electric field ${\vec{E}}$ is given by ${\vec{F}=q\vec{E}}$. From basic algebra we can deduce that the magnitude of the electric field from a source charge ${Q}$ is given by ${|\vec{E}|= \frac{kQ}{r^2}}$.

This generalizes to continuous distributions: ${E=\int \frac{kQ}{R^2} \hat{r}}$ and we utilize the same procedures for continuous charge distributions as we did for calculating the electric force.

Instead of using the ${k}$ constant, it turns out that there is a more fundamental constant ${\epsilon_0}$ known as permittivity. ${\epsilon_0}$ and ${k}$ are related by ${\frac{1}{4 \pi \epsilon_0}=k}$.