# Is the potential difference independent of the charge

## Work in the electric field

The electric field of a point charge

Radial lines: Field (force) lines

Circles (or spherical surfaces): equipotential surfaces

The field lines are perpendicular to the equipotential surfaces.

The definite path integral over the electric field is called tension U:

As Electric field potential V (x, y, z) is called the indefinite integral

With the help of the potential one can tension as Potential difference express:

The potential is (like every indefinite integral) determined up to a constant. This constant is determined by practical considerations. As a rule, this constant is chosen in such a way that the potential at infinity or on the ground becomes zero. This definition has no effect on the amount of the potential difference.

If one knows the potential of an electric field, the work in the electric field can be expressed very simply by the voltage (potential difference) U:

Potential energy is the work that has to be done to bring a charge from a given point in space to a place with zero potential. Where the point with zero potential is located is determined by definition because of the freedom of choice of the constant of integration C. With

Potential energy and potential have opposite signs. This also applies:

A field has a unique potential if the work to shift the charge is path-independent. This means that

If the potential of a field depends only on the location, then the work done in a closed way is zero.

In this case, the electric field can also be clearly derived from the potential energy and the following applies:

The following applies to the total mechanical energy:

If one sets the constant C = 0, the potential disappears at infinity. With this calibration one obtains q for the Coulomb potential of the charge

The potential energy of a charge q ’in the field of charge q is then obtained by means of

For an electron (q = -e) in the field of a proton (q ’= + e), both the Coulomb potential and the potential energy are negative.

### The electric flux F is understood to be a quantity that is proportional to the field generated by electric charges and the area traversed by this field:

or.

In a sense, the field strength is a measure of the density of the electric field lines and the flux is a measure of the total number of field lines penetrating an area. The scalar product takes into account the orientation of the surface to the field. If is directed parallel to the surface, the field lines do not penetrate the surface. Consequently the flow would be zero. The integral takes into account the change in the field (amount and direction) over the area.

### Gauss - Ostrogradski's theorem

The total flow through any closed area is proportional to the total charge Q within that area: