Electric fields are caused by
electric charges, described by
Gauss's law, and time varying
magnetic fields, described by
Faraday's law of induction. Together, these laws are enough to define the behavior of the electric field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form
Maxwell's equations that describe both fields as a function of charges and
currents. s 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.
Electrostatics In the special case of a
steady state (stationary charges and currents), the Maxwell-Faraday
inductive effect disappears. The resulting two equations (Gauss's law \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} and Faraday's law with no induction term \nabla \times \mathbf{E} = 0), taken together, are equivalent to
Coulomb's law, which states that a particle with electric charge q_1 at position \mathbf r_1 exerts a force on a particle with charge q_0 at position \mathbf r_0 of: \mathbf{F}_{01} = \frac{q_1q_0}{4\pi\varepsilon_0} {\hat\mathbf r_{01}\over ^2} = \frac{q_1q_0}{4\pi\varepsilon_0} {\mathbf r_{01}\over ^3} where • \mathbf{F}_{01} is the force on charged particle q_0 caused by charged particle q_1 . • is the
permittivity of free space. • \hat \mathbf{r}_{01} is a
unit vector directed from \mathbf r_1 to \mathbf r_0 . • \mathbf{r}_{01} is the
displacement vector from \mathbf r_1 to \mathbf r_0 . Note that \varepsilon_0 must be replaced with \varepsilon,
permittivity, when charges are in non-empty media. 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 \mathbf r_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the
source charge) \mathbf{E}_{1} (\mathbf r_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1}{4\pi\varepsilon_0} {\hat\mathbf r_{01}\over ^2} = \frac{q_1}{4\pi\varepsilon_0} {\mathbf r_{01}\over ^3} where \mathbf{E}_{1} (\mathbf r_0) is the component of the electric field at q_0 due to q_1 . This is the
electric field at point \mathbf r_0 due to the point charge q_1; it is a
vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf r_0. Since this formula gives the electric field magnitude and direction at any point \mathbf r_0 in space (except at the location of the charge itself, \mathbf r_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. 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 \mathbf{F} = q\mathbf{E} . The
SI unit of the electric field is the
newton per
coulomb (N/C), or
volt per
meter (V/m); in terms of the
SI base units it is kg⋅m⋅s−3⋅A−1.
Superposition principle Due to the
linearity of
Maxwell's equations, electric fields satisfy the
superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. This is analogous to the
gravitational potential. The difference between the electric potential at two points in space is called the
potential difference (or voltage) between the two points. In general, however, the electric field cannot be described independently of the magnetic field. Given the
magnetic vector potential, , defined so that {{tmath|1= \mathbf{B} = \nabla \times \mathbf{A} }}, one can still define an electric potential \varphi such that: \mathbf{E} = - \nabla \varphi - \frac { \partial \mathbf{A} } { \partial t } , where \nabla \varphi is the
gradient of the electric potential and \frac { \partial \mathbf{A} } { \partial t } is the
partial derivative of with respect to time.
Faraday's law of induction can be recovered by taking the
curl of that equation \nabla \times \mathbf{E} = -\frac{\partial (\nabla \times \mathbf{A})} {\partial t} = -\frac{\partial \mathbf{B}} {\partial t} , which justifies, a posteriori, the previous form for .
Continuous vs. discrete charge representation The equations of electromagnetism are best described in a continuous description. However, charges are sometimes best described as discrete points; for example, some models may describe
electrons as point sources where charge density is infinite on an infinitesimal section of space. A charge q located at \mathbf{r}_0 can be described mathematically as a charge density {{tmath|1= \rho(\mathbf{r}) = q\delta(\mathbf{r} - \mathbf{r}_0)}}, where the
Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges. == Electrostatic fields ==