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Electric field work is the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
performed by an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
on a charged particle in its vicinity. The particle located experiences an interaction with the electric field. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
at those points. The work can be done, for example, by electrochemical devices (
electrochemical cell An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or using electrical energy to cause chemical reactions. The electrochemical cells which generate an electric current are called voltaic o ...
s) or different metals junctions generating an
electromotive force In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
. Electric field work is formally equivalent to work by other force fields in physics, and the formalism for electrical work is identical to that of mechanical work.


Physical process

Particles that are free to move, if positively charged, normally tend towards regions of lower electric potential (net negative charge), while negatively charged particles tend to shift towards regions of higher potential (net positive charge). Any movement of a positive charge into a region of higher potential requires external work to be done against the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
, which is equal to the work that the electric field would do in moving that positive charge the same distance in the opposite direction. Similarly, it requires positive external work to transfer a negatively charged particle from a region of higher potential to a region of lower potential.
Kirchhoff's voltage law Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
, one of the most fundamental laws governing electrical and electronic circuits, tells us that the voltage gains and the drops in any electrical circuit always sum to zero. The formalism for electric work has an equivalent format to that of mechanical work. The work per unit of charge, when moving a negligible test charge between two points, is defined as the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
between those points. : W = Q \int_^ \mathbf \cdot \, d \mathbf = Q \int_^ \frac \cdot \, d \mathbf= \int_^ \mathbf \cdot \, d \mathbf where :''Q'' is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
of the particle :''E'' is the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
, which at a location is the force at that location divided by a unit ('test') charge :''F''E is the
Coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
(electric) force :''r'' is the
displacement Displacement may refer to: Physical sciences Mathematics and Physics * Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
:''\cdot'' is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
operator


Mathematical description

Given a charged object in empty space, Q+. To move q+ ''closer'' to Q+ (starting from r_0 = \infty , where the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
=0, for convenience), we would have to apply an external force against the Coulomb field and positive work would be performed. Mathematically, using the definition of a
conservative force In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum ...
, we know that we can relate this force to a
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
gradient as: :-\frac = \mathbf_ Where U(r) is the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of q+ at a distance r from the source Q. So, integrating and using
Coulomb's Law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
for the force: :U(r) = \Delta U = -\int_^ \mathbf_ \cdot \, d \mathbf= -\int_^ \frac\frac \cdot \, d \mathbf= \frac\left(\frac- \frac\right) = -\frac \frac Now, use the relationship : W = -\Delta U \! To show that the external work done to move a point charge q+ from infinity to a distance r is: :W_ = \frac\frac This could have been obtained equally by using the definition of W and integrating F with respect to r, which will ''prove'' the above relationship. In the example both charges are positive; this equation is applicable to any charge configuration (as the product of the charges will be either positive or negative according to their (dis)similarity). If one of the charges were to be negative in the earlier example, the work taken to wrench that charge away to infinity would be exactly the same as the work needed in the earlier example to push that charge back to that same position. This is easy to see mathematically, as reversing the boundaries of integration reverses the sign.


Uniform electric field

Where the electric field is constant (i.e. ''not'' a function of displacement, r), the work equation simplifies to: :W = Q (\mathbf \cdot \, \mathbf)=\mathbf \cdot \, \mathbf or 'force times distance' (times the cosine of the angle between them).


Electric power

The
electric power Electric power is the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt, one joule per second. Standard prefixes apply to watts as with other SI units: thousands, millions and billions o ...
is the rate of energy transferred in an electric circuit. As a partial derivative, it is expressed as the change of work over time: :P=\frac=\frac, where V is the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
. Work is defined by: : \delta W = \mathbf\cdot\mathbf\delta t, Therefore :\frac=\mathbf \cdot \,\mathbf


References

{{Reflist Electromagnetism