## Definition

From Coulomb's law a particle with electric charge $q_{1}$ at position ${\boldsymbol {x}}_{1}$ exerts a force on a particle with charge $q_{0}$ at position ${\boldsymbol {x}}_{0}$ of

- ${\boldsymbol {F}}={1 \over 4\pi \varepsilon _{0}}{q_{1}q_{0} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{0})^{2}}{\hat {\boldsymbol {r}}}_{1,0}$
- where ${\boldsymbol {r}}_{1,0}$ is the unit vector in the direction from point ${\boldsymbol {x}}_{1}$ to point ${\boldsymbol {x}}_{0}$, and
*ε*_{0} is the electric constant (also known as "the absolute permittivity of free space") in C^{2} m^{−2} N^{−1}

When the charges $q_{0}$ and $q_{1}$ have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other. When the charges have unlike signs the force is negative, indicating the particles attract.
To make it easy to calculate the Coulomb force on any charge at position ${\boldsymbol {x}}_{0}$ this expression can be divided by $q_{0}$, leaving an expression that only depends on the other charge (the *source* charge)^{[8]}^{[6]}

- ${\boldsymbol {E}}({\boldsymbol {x}}_{0})={{\boldsymbol {F}} \over q_{0}}={1 \over 4\pi \varepsilon _{0}}{q_{1} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{0})^{2}}{\hat {\boldsymbol {r}}}_{1,0}$

This is the *electric field* at point ${\boldsymbol {x}}_{0}$ due to the point charge $q_{1}$; it is a vector equal to the Coulomb force per unit charge that a positive point charge would experience at the position ${\boldsymbol {x}}_{0}$.
Since this formula gives the electric field magnitude and direction at any point ${\boldsymbol {x}}_{0}$ in space (except at the location of the charge itself, ${\boldsymbol {x}}_{1}$, where it becomes infinite) it defines a vector field.
From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge.

If there are multiple charges, the resultant Coulomb force on a charge can be found by summing the vectors of the forces due to each charge. This shows the electric field obeys the *superposition principle*: the total electric field at a point due to a collection of charges is just equal to the vector sum of the electric fields at that point due to the individual charges.^{[6]}^{[9]}

- ${\boldsymbol {E}}({\boldsymbol {x}})={\boldsymbol {E}}_{1}({\boldsymbol {x}})+{\boldsymbol {E}}_{2}({\boldsymbol {x}})+{\boldsymbol {E}}_{3}({\boldsymbol {x}})+\cdots ={1 \over 4\pi \varepsilon _{0}}{q_{1} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{1}+{1 \over 4\pi \varepsilon _{0}}{q_{2} \over ({\boldsymbol {x}}_{2}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{2}+{1 \over 4\pi \varepsilon _{0}}{q_{3} \over ({\boldsymbol {x}}_{3}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{3}+\cdots$
- ${\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}\sum _{k=1}^{N}{q_{k} \over ({\boldsymbol {x}}_{k}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{k}$
- where ${\boldsymbol {{\hat {r}}_{k}}}$ is the unit vector in the direction from point ${\boldsymbol {x}}_{k}$ to point ${\boldsymbol {x}}$.

This is the definition of the electric field due to the point *source charges* $q_{1},\ldots ,q_{N}$.
It diverges and becomes infinite at the locations of the charges themselves, and so is not defined there.

The Coulomb force on a charge of magnitude $q$ at any point in space is equal to the product of the charge and the electric field at that point

- ${\boldsymbol {F}}=q{\boldsymbol {E}}$

The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m); in terms of the SI base units they are kg⋅m⋅s^{−3}⋅A^{−1}

The electric field due to a continuous distribution of charge $\rho ({\boldsymbol {x}})$ in space (where $\rho$ is the charge density in coulombs per cubic meter) can be calculated by considering the charge $\rho ({\boldsymbol {x}}')dV$ in each small volume of space $dV$ at point ${\boldsymbol {x}}'$ as a point charge, and calculating its electric field $d{\boldsymbol {E}}({\boldsymbol {x}})$ at point ${\boldsymbol {x}}$

- $d{\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}{\rho ({\boldsymbol {x}}')dV \over ({\boldsymbol {x}}'-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}'$

where ${\hat {\boldsymbol {r}}}'$ is the unit vector pointing from ${\boldsymbol {x}}'$ to ${\boldsymbol {x}}$, then adding up the contributions from all the increments of volume by integrating over the volume of the charge distribution $V$

- ${\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}\iiint \limits _{V}\,{\rho ({\boldsymbol {x}}')dV \over ({\boldsymbol {x}}'-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}'$