# Work (electric field)

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Electric field work is the work performed by an electric field on a charged particle in its vicinity.

A charged particle located within the influence of an electric field experiences an interaction that is formally equivalent to other work by force fields in physics. The electric field performs work on the particle.[1] 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 at those points. The work can be done, for example, by electrochemical devices (electrochemical cells) or different metals junctions generating an electromotive force. The physical and mathematical 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, 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.

The electric force is a conservative force: work done by a static electric field is independent of the path taken by the charge. There is no change in the electric potential around any closed path; when returning to the starting point in a closed path, the net of the external work done is zero. The same holds for electric fields.

This is the basis of Kirchhoff's voltage law, one of the most fundamental laws governing electrical and electronic circuits, according to which 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 between those points.

${\displaystyle W=Q\int _{a}^{b}\mathbf {E} \cdot \,d\mathbf {r} =Q\int _{a}^{b}{\frac {\mathbf {F_{E}} }{Q}}\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F_{E}} \cdot \,d\mathbf {r} }$

where

Q is the electric charge of the particle, q, the unit charge
E is the electric field, which at a location is the force at that location divided by a unit ('test') charge
FE is the Coulomb (electric) force
r is the displacement
${\displaystyle \cdot }$  is the dot product

## Mathematical description

Given a charged object in empty space, Q+. To move q+ (with the same charge) closer to Q+ (starting from infinity, where the potential energy=0, for convenience), positive work would be performed. Mathematically:

${\displaystyle -{\frac {\partial U}{\partial \mathbf {r} }}=\mathbf {F} }$

In this case, U is the potential energy of q+. So, integrating and using Coulomb's Law for the force:

${\displaystyle U=-\int _{r_{0}}^{r}\mathbf {F} \cdot \,d\mathbf {r} =-\int _{r_{0}}^{r}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{\mathbf {r^{2}} }}\cdot \,d\mathbf {r} ={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}\left({\frac {1}{r_{0}}}-{\frac {1}{r}}\right)+c}$

c is usually set to 0 and r(0) to infinity (making the 1/r(0) term=0) Now, use the relationship

${\displaystyle W=-\Delta U\!}$

To show that in this case if we start at infinity and move the charge to r,

${\displaystyle W={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{\frac {1}{r}}}$

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:

${\displaystyle W=Q(\mathbf {E} \cdot \,\mathbf {r} )=\mathbf {F_{E}} \cdot \,\mathbf {r} }$

or 'force times distance' (times the cosine of the angle between them).

## Electric power

The electric power is the rate of energy transferred in an electric circuit. As a partial derivative, it is expressed as the change of work over time:

${\displaystyle P={\frac {\partial W}{\partial t}}={\frac {\partial QV}{\partial t}}}$ ,

where V is the voltage. Work is defined by:

${\displaystyle \delta W=\mathbf {F} \cdot \mathbf {v} \delta t,}$

Therefore

${\displaystyle {\frac {\partial W}{\partial t}}=\mathbf {F_{E}} \cdot \,\mathbf {v} }$

## References

1. ^ Debora M. Katz (1 January 2016). Physics for Scientists and Engineers: Foundations and Connections. Cengage Learning. pp. 1088–. ISBN 978-1-337-02634-5.