Electric field
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In physics, an electric field or E-field is an effect produced by an electric charge (or a time-varying magnetic field) that exerts a force on charged objects in the field. The SI units of the electric field are newtons per coulomb or volts per meter (both are equivalent). Electric fields contain electrical energy with energy density proportional to the square of the field intensity. Electric fields exist around all charges; the direction of field lines at a point is defined by the direction of the electric force exerted on a positive test charge placed at that point. The strength of the field is defined by the ratio of the electric force on a charge at a point to the magnitude of the charge placed at that point. In the dynamic case the electric field is accompanied by a magnetic field, by a flow of energy, and by real photons.
The concept of electric field was introduced by Michael Faraday.
The electric field or electric field intensity is a vector quantity, and the electric field strength is the magnitude of this vector.
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Definition and derivation (for electrostatics)
Electric field is defined as the electric force per unit charge. The direction of the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially outward from a positive charge and radially in toward a negative point charge.
The mathematical definition of the electric field is developed as follows. Coulomb's law gives the force between two point charges (infinitesimally small charged objects) as
- <math>
\mathbf{F} = \frac{1}{4 \pi \epsilon_0}\frac{q_1 q_2}{r^2}\mathbf{\hat r}\qquad(1) </math>
where
- <math>\epsilon_0</math> (pronounced epsilon-nought) is a physical constant, the permittivity of free space;
- <math>q_1</math> and <math>q_2</math> are the electric charges of the objects;
- <math>r</math> is the magnitude of the separation vector between the objects;
- <math>\hat r</math> is the unit vector representing the direction from the object exerting the force to the object experiencing the force. Thus, if <math>q_1</math> and <math>q_2</math> have the same sign, then the force on each object is in the direction away from the other.
In the SI system of units, force is given in newtons, charge in coulombs, and distance in metres. Thus, <math>\epsilon_0</math> has units of C²/(N·m²).
This was known empirically. Suppose one of the charges is taken to be fixed, and the other one to be a moveable "test charge". Note that according to this equation, the force on the test object is proportional to its charge. The electric field is defined as the proportionality constant between charge and force (in other words, the force per unit of test charge):
- <math>
\mathbf{F} = q\mathbf{E} </math>
- <math>
\mathbf{E} = \frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\mathbf{\hat r} </math>
However, note that this equation is only true in the case of electrostatics, that is to say, when there is nothing moving. The more general case of moving charges causes this equation to become the Lorentz force equation. When we speak of a "moveable test charge", this means only that the above equations hold regardless of the position of the (stationary) test charge.
Furthermore, 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.
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.
- <math>\mathbf{E}_{\rm tot} = \sum_i \mathbf{E}_i = \mathbf{E}_1 + \mathbf{E}_2 + \mathbf{E
}_3 \ldots \,\!</math>
If this principle is extended to an infinite number of infinitesimally small elements of charge, the following formula results:
- <math>
\mathbf{E} = \frac{1}{4\pi\epsilon_0} \int\frac{\rho}{r^2} \mathbf{\hat r}\,d^{3}\mathbf{r} </math>
where <math>\rho</math> 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,
- <math>
\mathbf{E} = -\mathbf{\nabla}\phi </math>
Where <math>\phi(x, y, z)</math> 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 <math>\varepsilon</math> of a material, which is the product of the permittivity of free space <math>\varepsilon_{0}</math> and the material-dependent relative permittivity <math>\varepsilon_{r}</math>, yields the Electric displacement field:
- <math>\mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_{0} \varepsilon_{r} \mathbf{E}</math>
Parallels between electrostatics and gravity
As explained above, electric field can be thought of as a proportionality constant when the force exerted on a test charge is proportional to the magnitude of the test charge. Put more simply, this is to say that the electrostatic environment affecting a point in space can be quantified by the electric field at that point; different physical facts of the environment combine to form this single vector quantity, and it is possible for different environments to produce the same vector quantity for electric field. Any given object (that we are measuring the force on) has associated various "weights;" the electrostatic weight is the charge, and the gravitic weight is the mass. The electrostatic force on some object in the environment is then simply the strength of the environment (the electric field), times the magnitude of the electrostatic weight (the charge). This is similar to gravity, where any given environment has a gravitational acceleration, and the force on some object in that environment is simply the acceleration due to gravity (the environmental factor) times the mass of the object (the gravitic weight). For electrostatics, the factors that determine the electric field in an environment are:
- The magnitude of each nearby charge
- The square of the distance between the "center" of that charge and the object being measured, and
- Coulomb's constant.
The analogous factors for determining a gravitational field are:
- The mass of each nearby object
- The square of the distance between the center of mass of that object and the object being measured, and
- The Universal Gravitational Constant.
When measuring the force on a mass at sea level due to Earth's gravity, the first factor (mass of the nearby environment-determining object) is the mass of the Earth, while the second factor (square of the distance between the environment-determining object and the measured object) is the square of the Earth's radius.
The units of the electric field, newtons per coulomb, can thus by expressed as force per unit charge.
Time-varying fields
Charges are not the only sources of electric fields. According to Faraday's law of induction,
- <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
where <math>\nabla \times \mathbf{E}</math> indicates the curl of the electric field, and <math>-\frac{\partial \mathbf{B}} {\partial t}</math> represents the vector rate of decrease of magnetic flux density 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. In this case, Coulomb's law no longer provides a useful definition of electric field as given above. Instead, the more general Gauss's Law, along with Faraday's law, determines the electric field.
See also
- Maxwell's equations are the full set of equations governing electric fields.
- Electromagnetism
- Electrostatics
- Electrodynamics
- Magnetism
External links
- Learning by Simulations Interactive simulation of an electric field of up to four point charges
- Java simulations of electrostatics in 2-D and 3-Dbg:Електрично поле
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