Electric field

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In physics, an electric field is a field of force with a field strength equal to the force per unit charge at that point. Basically, it is a field in which a charge experiences a force. The concept of an electric field was introduced by Michael Faraday.

The electric field is a with SI units of newtons per coulomb (N C−1) or, equivalently, volts per metre (V m−1). The SI base units of the electric field are kg·m·s−3·A−1. The strength of the field at a given point is defined as the force that would be exerted on a positive test charge of +1 coulomb placed at that point; the direction of the field is given by the direction of that force. Electric fields contain electrical energy with energy density proportional to the square of the field amplitude. The electric field is to charge as gravitational acceleration is to mass and force density is to volume.

An electric field that changes with time (such as due to the motion of charged particles in the field) will also influence the magnetic field of that region of space. Thus, in general, the electric and magnetic fields are not completely separate phenomena; what one observer perceives as an electric field, another observer in a different frame of reference perceives as a mixture of electric and magnetic fields. For this reason, one speaks of "electromagnetism" or "electromagnetic fields." In quantum mechanics, disturbances in the electromagnetic fields are called photons, and the energy of photons is quantized.

Contents

Definition

Something that surrounds a positive or negative charge

\mathbf{E} = \frac{\mathbf{F}}{q}

where

\mathbf{F} is the force acting on the charge.
q is its charge
\mathbf{E} is the magnitude of the electric field.

Taken literally, this equation only defines the electric field at the places where there are stationary charges present to experience it. Furthermore, the force exerted by another charge q will alter the source distribution, which means the electric field in the presence of q differs from itself in the absence of q. However, the electric field of a given source distribution remains defined in the absence of any charges with which to interact. This is achieved by measuring the force exerted on successively smaller test charges placed in the vicinity of the source distribution. By this process, the electric field created by a given source distribution is defined as the limit as the test charge approaches zero of the force per unit charge exerted thereupon.

\mathbf{E}=\lim_{q \to 0}\frac{\mathbf{F}}{q}

This allows the electric field to be dependent on the source distribution alone.

As is clear from the definition, the direction of the electric field is the same as the direction of the force it would exert on a positively-charged particle, and opposite the direction of the force on a negatively-charged particle. Since like charges repel and opposites attract (as quantified below), the electric field tends to point away from positive charges and towards negative charges.

Based on Coulomb's Law for interacting point charges, the contribution to the E-field at a point in space due to a single, discrete charge located at another point in space is given by the following:

\mathbf{E}= {1 \over 4\pi\varepsilon_0}{q \over r^2}\mathbf{\hat{r}} \

where

q is the charge of the particle creating the electric force,
r is the distance from the particle with charge q to the E-field evaluation point,
\mathbf{\hat{r}} is the unit vector pointing from the particle with charge q to the E-field evaluation point,
\varepsilon_0 is the electric constant.

The total E-field due to a quantity of point charges, nq, is simply the superposition of the contribution of each individual point charge:

\mathbf{E} = \sum_{i=1}^{n_q} {\mathbf{E}_i} = \sum_{i=1}^{n_q} {{1 \over 4\pi\varepsilon_0}{q_i \over r_i^2}\mathbf{\hat{r}}_i}

Alternately, Gauss's Law allows the E-field to be calculated in terms of a continuous distribution of charge density in space, ρ:

\nabla \cdot \mathbf{E} = \frac { \rho } { \varepsilon _0 }

Coulomb's law is actually a special case of Gauss's Law, a more fundamental description of the relationship between the distribution of electric charge in space and the resulting electric field. Gauss's law is one of Maxwell's equations, a set of four laws governing electromagnetics.

Time-varying fields

Charges do not only produce electric fields. As they move, they generate magnetic fields, and if the magnetic field changes, it generates electric fields. A changing magnetic field gives rise to an electric field,

\mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }

which yields Faraday's law of induction,

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}

where

\nabla \times \mathbf{E} indicates the curl of the electric field,
-\frac{\partial \mathbf{B}} {\partial t} represents the vector rate of decrease of magnetic field with time.

This means that a magnetic field changing in time produces a curled electric field, possibly also changing in time. The situation in which electric or magnetic fields change in time is no longer electrostatics, but rather electrodynamics or electromagnetics.

Properties (in electrostatics)

Illustration of the electric field surrounding a positive (red) and a negative (green) charge.

According to equation (1) above, electric field is dependent on position. The electric field due to any single charge falls off as the square of the distance from that charge.

Electric fields follow the superposition principle. If more than one charge is present, the total electric field at any point is equal to the vector sum of the respective electric fields that each object would create in the absence of the others.

\mathbf{E}_{\rm total} = \sum_i \mathbf{E}_i = \mathbf{E}_1 + \mathbf{E}_2 + \mathbf{E}_3 \ldots \,\!

If this principle is extended to an infinite number of infinitesimally small elements of charge, the following formula results:

\mathbf{E} = \frac{1}{4\pi\varepsilon_0} \int\frac{\rho}{r^2} \mathbf{\hat{r}}\,\mathrm{d}V

where

ρ is the charge density, or the amount of charge per unit volume.

The electric field at a point is equal to the negative gradient of the electric potential there. In symbols,

\mathbf{E} = -\nabla \phi

where

φ(x,y,z) is the scalar field representing the electric potential at a given point.

If several spatially distributed charges generate such an electric potential, e.g. in a solid, an electric field gradient may also be defined.

Considering the permittivity \varepsilon of a linear material, which may differ from the permittivity of free space \varepsilon_{0}, the electric displacement field is:

\mathbf{D} = \varepsilon \mathbf{E}

Energy in the electric field

Main article: Electrical energy

The electric field stores energy. The energy density of the electric field is given by

u = \frac{1}{2} \varepsilon |\mathbf{E}|^2 \, ,

where ε is the permittivity of the medium in which the field exists, and E is the electric field vector.

The total energy stored in the electric field in a given volume V is therefore

\frac{1}{2} \varepsilon \int_{V} |\mathbf{E}|^2 \, \mathrm{d}V \, ,

where dV is the differential volume element.

Parallels between electrostatics and gravity

Coulomb's law, which describes the interaction of electric charges:

\mathbf{F} = \frac{1}{4 \pi \varepsilon_0}\frac{Qq}{r^2}\mathbf{\hat{r}} = q\mathbf{E}

is similar to Newton's law of universal gravitation:

\mathbf{F} = G\frac{Mm}{r^2}\mathbf{\hat{r}} = m\mathbf{g}

This suggests similarities between the electric field E and the gravitational field g, so sometimes mass is called "gravitational charge".

Similarities between electrostatic and gravitational forces:

  1. Both act in a vacuum.
  2. Both are central and conservative.
  3. Both obey an inverse-square law (both are inversely proportional to square of r).
  4. Both propagate with finite speed c.

Differences between electrostatic and gravitational forces:

  1. Electrostatic forces are much greater than gravitational forces (by about 1036 times).
  2. Gravitational forces are attractive for like charges, whereas electrostatic forces are repulsive for like charges.
  3. There are no negative gravitational charges (no negative mass) while there are both positive and negative electric charges. This difference combined with previous implies that gravitational forces are always attractive, while electrostatic forces may be either attractive or repulsive.
  4. Electric charge is invariant while relativistic mass is not.

See also

External links

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This page was last modified on 10 March 2010 at 01:28.

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