Electric field

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.

Definition

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.

:$\backslash mathbf\{E\}=\backslash lim\_\{q\; \backslash to\; 0\}\backslash frac\{\backslash mathbf\{F\}\}\{q\}$

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:

:$\backslash mathbf\{E\}\; =\; \backslash sum\_\{i=1\}^\{n\_q\}\; \{\backslash mathbf\{E\}\_i\}\; =\; \backslash sum\_\{i=1\}^\{n\_q\}\_i\}.$

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

:$\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash frac\; \{\; \backslash rho\; \}\; \{\; \backslash varepsilon\; \_0\; \}.$

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

$E\; =\; -\; \backslash frac\{V\}\{d\}$

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

:$\backslash mathbf\{E\}\; =\; -\; \backslash nabla\; \backslash phi\; -\; \backslash frac\; \{\; \backslash partial\; \backslash mathbf\{A\}\; \}\; \{\; \backslash partial\; t\; \}$

where,

:$\backslash mathbf\{B\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}$

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,

:$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$

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

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 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.

:$\backslash mathbf\{E\}\_\{\backslash rm\; total\}\; =\; \backslash sum\_i\; \backslash mathbf\{E\}\_i\; =\; \backslash mathbf\{E\}\_1\; +\; \backslash mathbf\{E\}\_2\; +\; \backslash mathbf\{E\}\_3\; \backslash ldots\; \backslash ,\backslash !$

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

:$\backslash mathbf\{E\}\; =\; \backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}\; \backslash int\backslash frac\{\backslash rho\}\{r^2\}\; \backslash mathbf\{\backslash hat\{r\}\}\backslash ,\backslash mathrm\{d\}V$

where :$\backslash rho$ 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,

:$\backslash mathbf\{E\}\; =\; -\backslash nabla\; \backslash Phi$

where :$\backslash Phi(x,\; y,\; z)$ 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:

:$\backslash mathbf\{D\}\; =\; \backslash varepsilon\; \backslash mathbf\{E\}.$

Energy in the electric field

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

:$u\; =\; \backslash frac\{1\}\{2\}\; \backslash varepsilon\; |\backslash mathbf\{E\}|^2\; \backslash ,\; ,$

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 :$\backslash frac\{1\}\{2\}\; \backslash varepsilon\; \backslash int\_\{V\}\; |\backslash mathbf\{E\}|^2\; \backslash ,\; \backslash mathrm\{d\}V\; \backslash ,\; ,$ where dV is the differential volume element.

Parallels between electrostatics and gravity

:$\backslash mathbf\{F\}\; =\; \backslash frac\{1\}\{4\; \backslash pi\; \backslash varepsilon\_0\}\backslash frac\{Qq\}\{r^2\}\backslash mathbf\{\backslash hat\{r\}\}\; =\; q\backslash mathbf\{E\}$

is similar to Newton's law of universal gravitation:

:$\backslash mathbf\{F\}\; =\; G\backslash frac\{Mm\}\{r^2\}\backslash mathbf\{\backslash hat\{r\}\}\; =\; m\backslash mathbf\{g\}.$

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: # Both act in a vacuum. # Both are central and conservative. # Both obey an inverse-square law (both are inversely proportional to square of r). # Both propagate with finite speed c, the speed of light. # Electric charge and relativistic mass are conserved; note, though, that rest mass is not conserved.

Differences between electrostatic and gravitational forces: # Electrostatic forces are much greater than gravitational forces (by about 10³⁶ times). # Gravitational forces are attractive for like charges, whereas electrostatic forces are repulsive for like charges. # 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

External links

Category:Electrostatics Category:Physical quantities Category:Introductory physics Category:Fundamental physics concepts Category:Electromagnetism