Electric field

In physics, an electric field E is a vector field in which each point is associated with a vector representing a force F acting on an electric charge q,

$\mathbf {F} =q\mathbf {E} .$

Often E is due to the presence in its neighborhood of one or more electric charges other than q, but E may also be caused by a magnetic field that varies in time, or by a combination of the two causes.

The direction of E is such that a positive charge q is pushed in the direction of the field vector (F and E parallel) and a negative charge q is pulled against the direction of E (F and E antiparallel).

The length |E| of E at a certain point is the strength of the electric field in that point, also known as the field intensity. The strength |E| ≡ E may be defined as the magnitude F ≡ |F| of the electric force exerted on a unit positive electric test charge q, or for arbitrary q by

$|\mathbf {F} |=q|\mathbf {E} |\quad \Longrightarrow \quad E={\frac {F}{q}}.$

The strength of the electric field does not depend on the test charge q. Strictly speaking, the introduction of a small test charge, which itself causes an electric field, slightly modifies the existing field. The electric field may therefore be defined as the force per positive charge Δq that is so small that the field can be assumed undisturbed by the presence of Δq. The strength of the electric field due to a single point charge is given by Coulomb's law.

An electric field may be time-dependent, as in the case of a field caused by charges accelerating up and down the transmitting antenna of a television station. Such a field is always accompanied by a magnetic field. The electric field with an accompanying magnetic field is propagated through space as an electromagnetic wave at the same speed as that of light.

When there is no time-dependent magnetic field present, the electric field E is related to an electric potential Φ,

$\mathbf {E} \equiv (E_{x},\;E_{y},\;E_{z})=(-{\frac {\partial \Phi }{\partial x}},\;-{\frac {\partial \Phi }{\partial y}},\;-{\frac {\partial \Phi }{\partial z}})\quad {\hbox{or}}\quad \mathbf {E} =-{\boldsymbol {\nabla }}\Phi .$

The electric field has dimension force per charge or, equivalently, voltage per length. In the SI system, the appropriate units are newton per coulomb, equivalent to volt per meter. In Gaussian units, the electric field is expressed in units of dyne per statcoulomb (formerly known as esu), equivalent to statvolt per centimeter.

Mathematical description

An electric field E may be due to the presence of charges by Gauss's law, which in differential form is one of the microscopic Maxwell equations

${\boldsymbol {\nabla }}\cdot \mathbf {E} ={\frac {\rho (\mathbf {r} )}{\epsilon _{0}}},$

where ε₀ is the electric constant, ρ(r) is a charge distribution, and ∇· stands for the divergence of E.

The field may also be caused by a varying magnetic field as shown by Faraday's law (one of Maxwell's equations),

${\boldsymbol {\nabla }}\times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\equiv -{\dot {\mathbf {B} }},$

where B is the magnetic flux density (also known as magnetic induction), and the symbol ∇× stands for the curl of E.

Because of the Helmholtz decomposition of a general vector field, we can write

$\mathbf {E} ={\boldsymbol {\nabla }}\times \mathbf {C} (\mathbf {r} )-{\boldsymbol {\nabla }}\Phi (\mathbf {r} )$

with

$\Phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} ',$

which is the instantaneous (non-retarded) Coulomb potential due to ρ(r), and

$\mathbf {C} (\mathbf {r} )=-{\frac {1}{4\pi }}\int {\frac {\dot {\mathbf {B} }}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r} '.$

The field C(r) is related to the time derivative of the vector potential A if we require the Coulomb gauge. We introduce A and the Coulomb gauge, respectively:

$\mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} ,\qquad {\boldsymbol {\nabla }}\cdot \mathbf {A} =0.$

One can then show that

${\boldsymbol {\nabla }}\times \mathbf {C} (\mathbf {r} )=-{\dot {\mathbf {A} }}(\mathbf {r} ).$

Hence the Helmholtz decomposition of the electric field (together with Coulomb gauge) gives the general expression for the electric field

$\mathbf {E} (\mathbf {r} )=-{\dot {\mathbf {A} }}(\mathbf {r} )-{\boldsymbol {\nabla }}\Phi (\mathbf {r} ).$

Clearly, when the time derivative of A vanishes, the field E is minus the gradient of the electric potential Φ.