Electrical work

Electrical work is the work done on a charged particle by an electric field. The equation for 'electrical' work is equivalent to that of 'mechanical' work:

where

The electrical work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points.

Particles that are free to move, if positively charged, normally tend towards regions of lower voltage (net negative charge), while if negatively charged they tend to shift towards regions of higher voltage (net positive charge).

However, any movement of a positive charge into a region of higher voltage requires external work to be done against the field of the electric force, work equal to that 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 voltage to a region of lower voltage.

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 voltage (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.

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:

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

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

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

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