Electric field

Electromagnetism
Electrostatics

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding. The concept of an electric field was introduced by Michael Faraday.

The electric field is a vector field 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 or magnitude 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, influences the local magnetic field. That is, 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 electrodynamics, disturbances in the electromagnetic fields are called photons, and the energy of photons is quantized.

Contents

Definition

The electric field intensity is defined as the force per unit positive charge that would be experienced by a stationary point charge, or "test charge", at a given location in the field:[1]

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

where

F is the electric force experienced by the test particle
qt is the charge of the test particle in the electric field
E is the electric field wherein the particle is located.

Taken literally, this equation only defines the electric field at a specific location as the force experienced by a stationary test charge at that point(with the sign of qt, positive or negative, determining the direction of the force). Given that electric fields are generated by electrically charged particles, adding and/or moving a source charge, qs, will alter the electric field distribution. Therefore, it is important to remember that an electric field is defined with respect to a particular configuration of source charges. In practice, this is achieved by placing test particles with successively smaller electric charge in the vicinity of the source distribution and measuring the force exerted on the test charges as their charge approaches zero.

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

This allows the electric field to be determined from the distribution of its source charges 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[1]:
\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,
ε0 is the electric constant.

The total E-field due to a quantity of point charges, n_q, is simply the superposition of the contribution of each individual point charge[2]:

\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}.

Alternatively, Gauss's law allows the E-field to be calculated in terms of a continuous distribution of charge density in space, ρ:[3]

\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.

Uniform fields

A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to each other and maintaining a voltage (potential difference) between them; it is only an approximation because of edge effects. Ignoring such effects, the equation for the magnitude of the electric field is:

E = - \frac{V}{d}

where

V is the voltage difference between the plates
d is the distance separating the plates

The negative sign arises as positive charges repel, so a positive charge will experience a force away from the positively charged plate, in the opposite direction to that in which the voltage increases.

Time-varying fields

An electric field can be produced, not only by a static charge, but also by a changing magnetic field. The combined electric field is expressed as,

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

where,

\mathbf{B} = \nabla \times \mathbf{A}

The vector B is the magnetic flux density and the vector A is the magnetic vector potential. Taking the curl of the electric field equation we obtain,

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

which is one of Maxwell's equations, referred to as Faraday's law of induction.[4]

Where electrostatics is the study of the fields surrounding static charges, the study of the electric fields induced by changing magnetic field comes under the domain of electrodynamics or electromagnetics.

Properties (in electrostatics)

According to Coulomb's law the 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 %2B \mathbf{E}_2 %2B \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

\rho 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

\Phi(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 ε of a linear material, which may differ from the permittivity of free space ε0, the electric displacement field is:

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

Energy in the electric field

Main article: Electric 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} = q(\frac{-1}{4 \pi \varepsilon_0}\frac{Q}{r^2}\mathbf{\hat{r})} = q\mathbf{E}

is similar to Newton's law of universal gravitation:

\mathbf{F} = m(-G\frac{M}{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, the speed of light.
  5. Electric charge and relativistic mass are conserved; note, though, that rest mass is not conserved.

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.

See also

References

  1. ^ a b Electric field in "Electricity and Magnetism", R Nave
  2. ^ 'The Electric Field' - Chapter 23 of Frank Wolfs's lectures at University of Rochester
  3. ^ 'Gauss's Law' - Chapter 24 of Frank Wolfs's lectures at University of Rochester
  4. ^ Huray, Paul G. (2009), Maxwell's Equations, Wiley-IEEE, p. 205, ISBN 0-470-54276-4, http://books.google.com/books?id=0QsDgdd0MhMCp , Chapter 7, p 205

External links