3.2. Work of electrostatic field forces

Let's find the formula for calculation of work, done by the forces of electrostatic field in the case of charge moving. Suppose the test charge q' is moving in the field of the point charge q from the point 1 to the point 2 (Fig. 3). As the force, which acts on the charge, is variable, all the path is needed to break on the little segments dl, in the frame of these segments the force can be constant. So, work dA on the segment we can define by the formula

. (7)

Work on the whole path 1 ̵ 2 equals to algebraic sum of work on the whole segments dl.

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The quantity ̵ is the projection of the segment dl on a field line. Using (2), we get

Fig. 3 – To the calculation of work of electrostatic force

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From the higher mathematics we know, that

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That's why

. (8)

As it follows from formula (8), the work of electrostatic field forces in the case of moving of the charge doesn't depend from the length and configuration of the path, but it depends from the initial and final position of the charge. Work on the closed path (r₁=r₂) equals zero. Forces, which have such properties and their fields, are called potential forces.

3.3. Field potential. Difference of potentials.

The charged body, which is brought in the potential electrostatic field, gets the potential energy. Forces of the field can move the charge in such way that its potential energy becomes decreasing. The work of these forces equals to the decrease of the potential energy.

(9)

In comparison with the formulas (8) and (9) we can see that potential energy of the charge q' in the field of the point charge q equals

_. (10)

Different test charges in the same point of the field get different energy, but relation of potential energy to the quantity of the charge is left constant. This relation is an energy characteristic of the field and called potential φ of the field in the given point.

Potential j in an electric field point – is a potential energy which the unit positive test charge possesses

_; . (11)

Substituting (10) in (11) we obtain potential of the field of the point charge q at the distance r far from it

. (12)

It is evident, that formula for work of electrostatic field forces (9) in the case of moving charge q' can be signed in as

. (13)

Whence potential difference =₁₂ between two points of 1 and 2 – is equal work done by an electrostatic force at transition of an unit positive test charge from one point of a field in another:

_; . (14)

The potential, as well as potential energy, is being defined relative to any zero value. Generally the potential can be accepted equal to zero in any point of a field. Usually the potential of infinitely distant points is taken as zero, and in the practical measurements – a ground potential.

If we remove charge q' from the given point of the field with the potential j₁=j to the point, which potential is zero j₂=0, the formula of work (13) rewrites like:

A₀=q'.

Then potential in a point can be considered as work done by force at transition of an unit positive test charge from the given point of a field j₁=j to a point j₂=0 where the potential is accepted equal to zero.

_; .

In a Si-system the potential and a potential difference is metered in Volts.

If potential difference between the points is equated 1 Volt, then transition of 1 Coulomb charge between these points requires 1 Joule of work of electric forces.

Potential is a scalar magnitude. Near a positive charge the potential always will be greater than zero, but near to negative charge - less than zero.

If the field is created by system of charges, potential of a resultant field in given point is equated to the algebraic sum of potentials from each charge in this point.