§ 7 Work of the electrostatic field intensity at moving charge.
Potential nature of the field forces.
Circulation of intensity
Consider the electrostatic field produced by the charge q. Let it move a test charge q_{0}. At any point of the field on the charge q_{0} a force
where force module ,  the unit vector of the radius vector , which determines the position of the charge q_{0 }relative to the charge q. Since the force varies from point to point, the work force of the electrostatic field can be written as a variable work force:
Because the
regarded charge transport from point 1 to point 2 along an arbitrary
trajectory, it can be concluded that the work on the movement of a
point charge in an electric field does not depend on the shape of the
path but is determined only the initial and final positions of the
charge. This indicates that the electrostatic field is potential, and
the strength of Coulomb  conservative force. Work on moving charge in a
field along a closed path is always zero.
 projection on the direction of the contour ℓ.
We take into account that the work on a closed path is zero
circulation of intensity .
Circulation of the electrostatic field, taken by an arbitrary closed path is always zero.
§ 7 Potential.
The link between the intensity and potential.
Gradient of potencial
Equipotential surfaces
Since the electrostatic field is a potential job of moving the charge in
such a field can be represented as a difference of the potential energy
of a charge in the start and end points of the path. (The work is equal
to the reduction of the potential energy, or the change in the
potential energy take with minus sign.)
Constant determined from the condition that the removal of the charge q_{0} to infinity the potential energy must be equal to zero.
.
Various test charges q_{0i}, placed at a given point of the field will have at this point various potential energies:
…
The ratio W_{pot} i to the value of the test charge q_{0i},
placed at a given point of the field is constant for a given point of
the field for all test charges. This ratio is called the potential.
Potential  energy characteristic of the electric field. Potential numerically equal to the potential energy, which has at a given point of the field unit positive charge.
Работу по перемещению заряда можно представить в виде
.
Potential is measured in volts
Equipotential surface is a surface of equal potential (φ = const). Work to move a charge along an equipotential surface is zero.
Relationship between the intensity and the potential φ can be found, based on the fact that the job of moving a charge q at the elementary segment dℓ can be written as
On the other hand
 gradient of potential .
Field intensity is equal to the potential gradient, to the negative.
Potential gradient shows how change the capacity per unit length. Gradient perpendicular to the function and
is directed towards the increase of the function. Therefore, the
vector intensity perpendicular to the equipotential surface and is
directed towards the decrease of the potential.
Consider the field created by a system N point charges q_{1}, q_{2}, … q_{N}. Distance from the charge to a given point of the field are r_{1}, r_{2}, … r_{N}. The work done by this field on the charge q_{0}, will be equal to the algebraic sum of the work force each charge separately.
where
The potential
field generated by a system of charges is defined as the algebraic
sum of the potentials produced at the same point each charge
separately.
§ 9 of difference potentials of the plane, the two planes, spheres, ball, cylinder
Using the relation between φ and оdefine the potential difference between two arbitrary points
The potential difference of the field of a uniformly charged infinite plane with surface charge density σ .
2. The potential difference of the field of two infinite parallel planes with an oppositely charged surface charge density σ .
If х_{1} = 0; х_{2} = d , then or
3. The potential difference of the field of a uniformly charged spherical surface of radius R .
If r_{1} = r, r_{2 }→ ∞, the potential outside the spheres
Inside a spherical surface potential everywhere and is equal
4. The potential difference of the field volume of a charged sphere of radius R and total charge Q .
Outside the ball r_{1}, r_{2} > R,
Inside the ball
5. The potential difference of the field of a uniformly charged cylinder (or infinitely long thread).
r > R:
