Equally interesting and no less important is the dipole field that arises under other circumstances. Let us have a body with complex distribution charge, say, like that of a water molecule (see Fig. 6.2), and we are only interested in the field far from it. We will show that it is possible to obtain a relatively simple expression for the fields, suitable for distances much larger than the dimensions of the body.
We can look at this body as an accumulation of point charges in a certain limited area (Fig. 6.7). (Later, if necessary, we will replace it with .) Let the charge be removed from the origin of coordinates, chosen somewhere within the group of charges, by a distance . What is the potential at a point located somewhere in the distance, at a distance much greater than the largest of ? The potential of our entire cluster is expressed by the formula
, (6.21)
where is the distance from to the charge (vector length ). If the distance from the charges to (to the observation point) is extremely large, then each of them can be taken as . Each term in the sum will become equal to , and can be taken out from under the sum sign. The result is simple
, (6.22)
where is the total charge of the body. Thus, we are convinced that from points sufficiently distant from the accumulation of charges, it appears to be just a point charge. This result is generally not very surprising.
Figure 6.7. Calculation of the potential at a point very distant from a group of charges.
But what if positive and negative charges will there be equal numbers in the group? The total charge will then be equal to zero. This is not such a rare case; we know that most bodies are neutral. The water molecule is neutral, but the charges in it are not located at one point, so that when we get close, we should notice some signs that the charges are separated. For the potential of an arbitrary charge distribution in a neutral body, we need an approximation that is better than that given by formula (6.22). Equation (6.21) is still valid, but it can no longer be assumed. A more precise expression is needed. To a good approximation, it can be considered different from (if the point is very distant) the projection of a vector onto a vector (see Fig. 6.7, but you should only imagine that it is much further away than shown). In other words, if - unit vector in the direction , then the next approximation to should be taken
But what we need is not, but; in our approximation (taking into account ) it is equal to
(6.24)
Substituting this into (6.21), we see that the potential is equal to
(6.25)
Ellipsis indicates members higher order by which we have neglected. Like the terms that we wrote out, these are subsequent terms of the Taylor series expansion in the neighborhood of powers of .
We have already obtained the first term in (6.25); in neutral bodies it disappears. The second term, like that of a dipole, depends on . Indeed, if we define
as a quantity describing charge distributions, then the second term of potential (6.25) turns into
i.e. just at dipole potential. The quantity is called the dipole moment of the distribution. This is a generalization of our previous definition; it reduces to it in the special case of point charges.
As a result, we found out that far enough from any set of charges the potential turns out to be dipole, as long as this set is generally neutral. It decreases as , and changes as , and its value depends on the dipole moment of the charge distribution. It is for this reason that dipole fields are important; pairs of point charges themselves are extremely rare.
For a water molecule, for example, dipole moment quite big. The electric field created by this moment is responsible for some important properties water. And for many molecules, say , the dipole moment disappears due to their symmetry. For such molecules, the decomposition must be carried out even more precisely, to the next terms of the potential, which decrease as is called the quadrupole potential. We will consider these cases later.
Equally interesting and no less important is the dipole field that arises under other circumstances. Let us have a body with a complex charge distribution, say, like a water molecule (see Fig. 6.2), and we are only interested in the field far from it. We will show that it is possible to obtain a relatively simple expression for the fields, suitable for distances much larger than the dimensions of the body.
We can look at this body as an accumulation of point charges q¡ in some limited area(Fig. 6.7). (Later, if necessary, we will replace q ¡ with ρdV.)
Let the charge q¡ be removed from the origin of coordinates, chosen somewhere within the group of charges, at a distance d¡. What is the potential at a point? R, located somewhere in the distance, at a distance R, much greater than the largest of d¡? The potential of our entire cluster is expressed by the formula
where r¡ is the distance from R to charge q¡
(length vector R-d¡). If the distance from the charges to R(up to the observation point) is extremely large, then each of r ¡ can be taken as R.
Each term will add up to q¡/R,
And 1/R
can be taken out from under the sum sign. The result is simple
Where Q is the total charge of the body. Thus, we are convinced that from points sufficiently distant from the accumulation of charges, it appears to be just a point charge. This result is generally not very surprising.
But what if there are equal numbers of positive and negative charges in the group? Total charge Q
then it will be equal to zero. This is not such a rare case; we know that most bodies are neutral. The water molecule is neutral, but the charges in it are not located at one point, so that when we get close, we should notice some signs that the charges are separated. For the potential of an arbitrary charge distribution in a neutral body, we need an approximation that is better than that given by formula (6.22). Equation (6.21) is still valid, but assume r¡ =R
no more. For r¡ I need a more precise expression. A good approximation r¡
can be considered different from R
(if point R very distant) onto the projection of vector d onto vector R (see Fig. 6.7, but you should just imagine that R much further than shown). In other words, if e r is a unit vector in the direction R, then for the next approach to r¡
need to accept
But we don't need r ¡
a 1/ r ¡
; in our approximation (taking into account d¡«R) it is equal to
Substituting this into (6.21), we see that the potential is equal to
The ellipsis indicates higher-order terms d/ R, which we have neglected. Like those terms that we wrote out, these are subsequent terms of expansion 1 / r¡ in a Taylor series in the neighborhood 1/R by degrees d¡/ R.
We have already obtained the first term in (6.25); in neutral bodies it disappears. The second term, like the dipole, depends on 1/R 2. Indeed, if we let's define
as a quantity describing charge distributions, then the second term of potential (6.25) turns into
i.e. just into the dipole potential. The value p is called dipole moment of the distribution. This is a generalization of our previous definition; it reduces to it in the special case of point charges.
In the end, we found out that it was quite far from any set of charges, the potential turns out to be dipole, as long as this set is generally neutral. It's decreasing like 1/ R 3 , and varies as cos θ, and its value depends on the dipole moment of the charge distribution. It is for this reason that dipole fields are important; pairs of point charges themselves are extremely rare.
A water molecule, for example, has a fairly large dipole moment. The electric field created by this moment is responsible for some important properties of water. And for many molecules, say CO 2, the dipole moment disappears due to their symmetry. For such molecules, the decomposition must be carried out even more precisely, to the next terms of the potential, decreasing as 1/ R 3 and called the quadrupole potential. We will consider these cases later.
A body located in a potential force field (electrostatic field) has potential energy, due to which work is done by the field forces. Job conservative forces occurs due to the loss of potential energy. Therefore, the work of electrostatic field forces can be represented as the difference in potential energies possessed by point charge Q 0 in initial and end points charge fields Q: , whence it follows that potential energy charge q 0 in the charge field Q equal to 
. It is determined ambiguously, but up to an arbitrary constant WITH. If we assume that when the charge is removed to infinity ( r®¥) potential energy vanishes ( U=0),
That WITH=0 and potential charge energy Q 0 ,
charge located in the field Q at a distance r from it, is equal to 
. For charges of the same name Q 0 Q> 0 and the potential energy of their interaction (repulsion) is positive, for unlike charges Q 0 Q<0 и потенциальная энергия их взаимодействия (притяжения) отрицательна.
Potential j at any point electrostatic field is a physical quantity determined by the potential energy of a unit positive charge placed at this point. From which it follows that the potential of the field created by a point charge Q, is equal to . Work done by electrostatic field forces when moving a charge Q 0 from point 1
to the point 2
, can be represented as , i.e., equal to the product of the moved charge and the potential difference at the initial and final points. Potential difference two points 1
And 2
in an electrostatic field is determined by the work done by field forces when moving a unit positive charge from a point 1
to the point 2
. Work done by field forces when moving a charge Q 0 from point 1
to the point 2
can also be written in the form 
. Expression for the potential difference: , where integration can be performed along any line connecting the starting and ending points, since the work of the electrostatic field forces does not depend on the trajectory of movement.
If you move the charge Q 0 from an arbitrary point beyond the field, i.e. to infinity, where, by condition, the potential is zero, then the work of the electrostatic field forces A ¥ =Q 0 j where
Potential- a physical quantity determined by the work of moving a single positive charge when it is removed from a given point in the field to infinity. This work is numerically equal to the work done by external forces (against the forces of the electrostatic field) to move a unit positive charge from infinity to a given point in the field. Unit of potential - volt(B): 1 V is the potential of a point in the field at which a charge of 1 C has a potential energy of 1 J (1 V = 1 J/C).
In the case of an electrostatic field, potential energy serves as a measure of the interaction of charges. Let there be a system of point charges in space Qi(i = 1, 2, ... ,n). The energy of everyone's interaction n charges will be determined by the relation
Where r ij - the distance between the corresponding charges, and the summation is carried out in such a way that the interaction between each pair of charges is taken into account once.
It follows from this that the field potential of the system of charges is equal to algebraic the sum of the field potentials of all these charges:
When considering the electric field created by a system of charges, the superposition principle should be used to determine the field potential:
The electric field potential of a system of charges at a given point in space is equal to the algebraic sum of the potentials of the electric fields created at a given point in space by each charge of the system separately:
6. Equipotential surfaces and their properties. Relationship between potential difference and electrostatic field strength.
An imaginary surface in which all points have the same potential is called an equipotential surface. The equation of this surface
If the field is created by a point charge, then its potential 
Thus, the equipotential surfaces in this case are concentric spheres. On the other hand, the tension lines in the case of a point charge are radial straight lines. Consequently, the tension lines in the case of a point charge perpendicular equipotential surfaces.
All points on the equipotential surface have the same potential, so the work done to move a charge along this surface is zero, i.e., the electrostatic forces acting on the charge are Always directed along the normals to equipotential surfaces. Therefore, the vector E always normal to equipotential surfaces, and therefore the vector lines E orthogonal to these surfaces.
An infinite number of equipotential surfaces can be drawn around each charge and each system of charges. However, they are usually carried out so that the potential differences between any two adjacent equipotential surfaces are the same. Then the density of equipotential surfaces clearly characterizes the field strength at different points. Where these surfaces are denser, the field strength is greater.
So, knowing the location of the electrostatic field strength lines, it is possible to construct equipotential surfaces and, conversely, from the known location of equipotential surfaces, the magnitude and direction of the field strength can be determined at each point in the field.
Let us find the relationship between the electrostatic field strength, which is its power characteristics, and potential - energy characteristic of the field.
Moving work single point positive charge from one point of the field to another along the axis X provided that the points are located infinitely close to each other and x 2 -x 1 = d x, equal to E x d x. The same work is equal j 1 -j 2 =dj. Equating both expressions, we can write
where the partial derivative symbol emphasizes that differentiation is performed only with respect to X. Repeating similar reasoning for the axes at And z, we can find the vector E:
Where i, j, k- unit vectors of coordinate axes x, y, z.
From the definition of gradient it follows that
i.e. tension E field is equal to the potential gradient with a minus sign. The minus sign is determined by the fact that the tension vector E fields directed to descending side potential.
To graphically depict the distribution of the electrostatic field potential, as in the case of the gravitational field, use equipotential surfaces- surfaces, at all points of which the potential j has the same meaning.
Field strength of a solitary positive point charge q at the point A at a distance r from the charge (Fig. 2.1) is equal to
Here is a unit vector directed along the straight line connecting this point and the charge.
Fig.2.1. Point charge field
Let the potential be zero at infinity. Then the potential of an arbitrary point in the field of a point charge
.
In the case of volumetric charge distribution (in a finite region), taking into account 
we have:
.
Similarly we have:
for surface charge distribution 
,
for linear charge distribution 
.
Poisson and Laplace equation
Previously received 
. Then:
From where we get the Poisson equation:
or 
.
- operator Laplace(Laplacian, delta operator).
In the Cartesian coordinate system 
can be presented in the form
Solution of Poisson's equation in general form can be found as follows. Let us assume that in volume V there are charges with density r. Let us represent these charges as a collection of point charges r dV, Where dV- volume element. Potential component d j electric field from elementary charge r dV equals 
.
The value of j is defined as the sum (integral) of the potentials from all field charges:
.
It is assumed that the potential at infinity is zero and the charges creating the fields are distributed in a limited area (otherwise the integral may turn out to be divergent).
In real conditions, free charges are located on the surface of conductors in an infinitely thin layer. In dielectrics that separate charged conductors, there are no space charges 
. In this case, in the dielectric we have the Laplace equation:
or 
.
To uniquely solve differential field equations, boundary conditions are required.
Boundary conditions for electric field vectors
Let a surface charge of density σ be distributed on the interface between two dielectrics with different dielectric constants ε 1 and ε 2.
Let us surround the point on the interface between the media with an elementary cylinder ( cylinder height much less than the radius) so that its bases are in different environments and are perpendicular to the normal drawn at the point in question (Fig. 2.2). This cylinder covers a small area on the interface between media with a charge σ.
The electric displacement vectors in the first and second media will be denoted by and , respectively.
Let us apply Gauss's theorem to the surface of the cylinder
,
Where S— surface of an elementary cylinder.
Fig.2.2. Vectors of electrical displacement at the boundary of media
Let us direct the volume of the cylinder to zero under the condition that the height of the cylinder is much less than its radius. In this case, the vector flow through the lateral surface can be neglected. Considering the small size of the base areas, we can assume that the vector within its area has the same value. Taking this into account, after integration for the projections of the vector onto the normal we obtain
Considering that 
, after reduction we obtain the boundary condition for the normal component of the electric displacement vector
Dn 2 –Dn 1 = σ . (**)
The normal projection of the electric displacement vector at the interface between two media undergoes a jump equal to the surface density of free charges distributed at this interface.
In the absence of a surface charge on the interface between the media, we have 
.
At the interface between two dielectrics, in the absence of a free charge at the interface between two media, the normal components of the electric displacement vector are equal.
Let us select a small contour at the interface between the media in such a way that its sides ab And CD were in different environments and were perpendicular to the normal drawn at the point in question (Fig. 2.3). The dimensions of the sides tend to zero; the contour satisfies the condition.
Fig.2.3. Vectors of electric field strength at the boundary of media
Let us apply Maxwell’s second equation in integral form to the contour:
,
where is the surface area limited by the contour abcd; is the vector of the elementary area directed perpendicular to the area.
When integrating, we neglect the contribution to the integral on the lateral sides da And bc due to their small size. Then:
Since the finite value tends to zero, then 
(***)
.
At the interface between two dielectrics, the tangential components of the electric field strength vector are equal.
If there is no surface charge on the interface between the media,
Expressions (*) and (***) we obtain a relationship that determines the refraction of vectors and at the interface between media
Formula - Coulomb's law
where k is the proportionality coefficient
q1,q2 stationary point charges
r distance between charges
3. Electric field strength- a vector physical quantity that characterizes the electric field at a given point and is numerically equal to the ratio of the force acting on a stationary test charge placed at a given point in the field to the magnitude of this charge: .
Electric field strength of a point charge
[edit] In SI units
For a point charge in electrostatics, Coulomb's law is true
Electric field strength of an arbitrary charge distribution
According to the principle of superposition for the field strength of a set of discrete sources, we have:
where is each
4. Superposition principle- one of the most general laws in many branches of physics. In its simplest formulation, the principle of superposition states:
· the result of the influence of several external forces on a particle is the vector sum of the influence of these forces.
The most famous principle of superposition is in electrostatics, in which it states that the strength of the electrostatic field created at a given point by a system of charges is the sum of the field strengths of individual charges.
The superposition principle can also take other formulations, which completely equivalent above:
· The interaction between two particles does not change when a third particle is introduced, which also interacts with the first two.
· The interaction energy of all particles in a many-particle system is simply the sum of the energies pair interactions between all possible pairs of particles. Not in the system many-particle interactions.
· The equations describing the behavior of a many-particle system are linear by the number of particles.
It is the linearity of the fundamental theory in the field of physics under consideration that is the reason for the emergence of the superposition principle in it.
In electrostatics The superposition principle is a consequence of the fact that Maxwell's equations in vacuum are linear. It follows from this that the potential energy of electrostatic interaction of a system of charges can be easily calculated by calculating the potential energy of each pair of charges.
5. Electric field work.
6. Electrostatic potential is equal to the ratio of the potential energy of interaction of a charge with a field to the magnitude of this charge:
The electrostatic field strength and potential are related by the relation
7. The principle of superposition of electrostatic fields. Forces or fields from different charges are added up taking into account their position or direction (vector). This expresses the principle of “superposition” of a field or potentials: the field potential of several charges is equal to the algebraic sum of the potentials of individual charges, φ=φ 1+φ2+…+φn= ∑i nφi. The sign of the potential coincides with the sign of the charge, φ=kq/r.
8. Potential energy of a charge in an electric field. Let us continue the comparison of the gravitational interaction of bodies and the electrostatic interaction of charges. Body mass m in the Earth's gravitational field has potential energy.
The work done by gravity is equal to the change in potential energy taken with the opposite sign:
A = -(W p2- W p1) = mgh.
(Hereinafter we will denote energy by the letter W.)
Just like a body of mass m in a gravity field has potential energy proportional to the mass of the body, an electric charge in an electrostatic field has potential energy W p, proportional to charge q. Work of electrostatic field forces A equal to the change in the potential energy of a charge in an electric field, taken with the opposite sign:
9. Theorem on the circulation of the tension vector in integral form: 
In differential form: 
10. The relationship between potential and tension. E= - grad = -Ñ .
The intensity at any point of the electric field is equal to the potential gradient at this point, taken with the opposite sign. The minus sign indicates that the tension E directed towards decreasing potential
11. Tension vector flow. 
Gauss's theorem in integral form: Where
· 
- flow of the electric field strength vector through a closed surface.
· - total charge contained in the volume that limits the surface.
· - electrical constant.
This expression represents Gauss's theorem in integral form.
In differential form: 
Here is the volumetric charge density (in the case of the presence of a medium, the total density of free and bound charges), and is the nabla operator.
12. Application of Gauss's law.1. Strength of the electrostatic field created uniformly charged spherical surface.
Let a spherical surface of radius R (Fig. 13.7) carry a uniformly distributed charge q, i.e. the surface charge density at any point on the sphere will be the same.
a. Let us enclose our spherical surface in a symmetrical surface S with radius r>R. The flux of the tension vector through the surface S will be equal to
By Gauss's theorem
Hence
c. Let's draw through point B, located inside the charged spherical surface, sphere S with radius r Field strength of a uniformly charged infinite rectilinear thread(or cylinder). Let us assume that a hollow cylindrical surface of radius R is charged with a constant linear density. Let us draw a coaxial cylindrical surface of radius. The flow of the tension vector through this surface By Gauss's theorem From the last two expressions we determine the field strength created by a uniformly charged thread: This expression does not include coordinates, therefore the electrostatic field will be uniform, and its intensity at any point in the field will be the same. 13. ELECTRIC DIPOLE. Electric dipole- a system of two equal in modulus opposite point charges (), the distance between which is significantly less than the distance to the field points under consideration. Dipole field potential: The dipole field strength along the extension of the dipole axis at the point A:
Dipole arm- vector directed along the dipole axis (a straight line passing through both charges) from a negative charge to a positive one and equal to the distance between the charges .
Electric dipole moment (dipole moment):
.
Dipole field strength at an arbitrary point (according to the principle of superposition):
where and are the field strengths created by positive and negative charges, respectively.
.
The field strength of a dipole at a perpendicular raised to the axis from its midpoint at the point B:
.
