P19-060 – Electric Field Created by a Point Charge

Electric field created by a point charge:

By symmetry, a point charge $q$ creates an electric field $\overrightarrow{E}$ everywhere in space surrounding that is radial. In the case of a positive point charge $q$, the electric field is radially outward and in the case of a negative point charge $-q$, the electric field is radially inward.

The formula of the electric field at a distance $r$ from the point charge $q$ is then given by

\boxed{\overrightarrow{E}=\frac{q}{4\pi {\varepsilon }_0r^2}\ \hat{r}}

where $\hat{r}$ is the unit vector that is oriented radially outward at that point as shown in the figure below.

The constant ${\varepsilon }_0$ is called the permittivity of free space and is a measure of the capability of an electric field to permeate a vacuum. It is equal to

\boxed{{\varepsilon }_0\approx 8.85\times {10}^{-12}\ A^2s^4m^{-3}kg^{-1}}

For simplicity, it can be thought of as a measure of how easily electric field lines can be established in a vacuum. The constant terms in the above formula are often grouped into a single constant $k$, called Coulomb’s constant, equal to

\boxed{k=\frac{1}{4\pi {\varepsilon }_0}\approx 9\times {10}^9\ \ N.m^2/C^2}

The formula for the electric field created by a point charge $q$ at a istance $r$ from the point charge is therefore often given in the form

\boxed{\overrightarrow{E}=\frac{kq}{r^2}\ \hat{r}}

Note: if instead of a positive point charge $q$ you consider a negative point charge $-q$, then the formula becomes

\overrightarrow{E}=-\frac{kq}{r^2}\ \hat{r}

where the negative sign can be interpreted as the direction of $\overrightarrow{E}$ being opposite that of $\hat{r}$ (i.e., radially inward instead of radially outward). Thus, the same formula conveniently applies for both positive and negative point charges.