# Electric field

Electric field emanating from a point positive electric charge suspended over an infinite sheet of conducting material.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them.[1][2] Electric field is sometimes abbreviated as E-field.[3] The electric field is defined mathematically as a vector field that associates to each point in space the (electrostatic or Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point.[4][5][6] The SI unit for electric field strength is volt per meter (V/m).[7] Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces (or interactions) of nature.

## Definition

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

${\displaystyle {\boldsymbol {F}}={1 \over 4\pi \varepsilon _{0}}{q_{1}q_{0} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{0})^{2}}{\hat {\boldsymbol {r}}}_{1,0}}$
where ${\displaystyle {\boldsymbol {r}}_{1,0}}$ is the unit vector in the direction from point ${\displaystyle {\boldsymbol {x}}_{1}}$ to point ${\displaystyle {\boldsymbol {x}}_{0}}$, and ε0 is the electric constant (also known as "the absolute permittivity of free space") in C2 m−2 N−1

When the charges ${\displaystyle q_{0}}$ and ${\displaystyle q_{1}}$ 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 ${\displaystyle {\boldsymbol {x}}_{0}}$ this expression can be divided by ${\displaystyle q_{0}}$, leaving an expression that only depends on the other charge (the source charge)[8][6]

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

This is the electric field at point ${\displaystyle {\boldsymbol {x}}_{0}}$ due to the point charge ${\displaystyle q_{1}}$; it is a vector equal to the Coulomb force per unit charge that a positive point charge would experience at the position ${\displaystyle {\boldsymbol {x}}_{0}}$. Since this formula gives the electric field magnitude and direction at any point ${\displaystyle {\boldsymbol {x}}_{0}}$ in space (except at the location of the charge itself, ${\displaystyle {\boldsymbol {x}}_{1}}$, 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]

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

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

Evidence of an electric field: styrofoam peanuts clinging to a cat's fur due to static electricity. The triboelectric effect causes an electrostatic charge to build up on the fur due to the cat's motions. The electric field of the charge causes polarization of the molecules of the styrofoam due to electrostatic induction, resulting in a slight attraction of the light plastic pieces to the charged fur. This effect is also the cause of static cling in clothes.

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

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

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

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

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

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