Let two point charges q 1 and q 2 be in vacuum at a distance r from each other. It can be shown that the potential energy of their interaction is given by the formula:

W = kq 1 q 2 /r (3)

We accept formula (3) without proof. Two features of this formula should be discussed.

First, where is the zero level of potential energy? After all, the potential energy, as can be seen from formula (3), cannot turn to zero. But in fact, the zero level exists, and it is at infinity. In other words, when the charges are located infinitely far from each other, the potential energy of their interaction is assumed to be zero (which is logical - in this case, the charges no longer "interact"). Secondly, q 1 and q 2 are again algebraic quantities of charges, i.e. charges according to their sign.

For example, the potential energy of the interaction of two similar charges will be positive. Why? If we let them go, they will start to accelerate and move away from each other.

Their kinetic energy increases, therefore, their potential energy decreases. But at infinity, the potential energy vanishes, and since it decreases to zero, it means that it is positive.

But the potential energy of interaction of opposite charges turns out to be negative. Indeed, let's remove them at a very large distance from each other - so that the potential energy is zero - and let go. The charges will begin to accelerate, approaching, and the potential energy decreases again. But if it was zero, then where should it decrease? Only towards negative values.

Formula (3) also helps to calculate the potential energy of a system of charges if the number of charges is more than two. To do this, you need to sum the energies of each pair of charges. We will not write out the general formula; Let us better illustrate what has been said with a simple example shown in Fig. eight

Rice. eight.

If the charges q 1, q 2, q 3 are at the vertices of a triangle with sides a, b, c, then the potential energy of their interaction is equal to:

W = kq 1 q 2 /a + kq 2 q 3 /b + kq 1 q 3 /c

Potential

From the formula W = - qEx we see that the potential energy of charge q in a uniform field is directly proportional to this charge. We see the same thing from the formula W \u003d kq 1 q 2 /r, the potential energy of a charge q 1 located in the field of a point charge q 2 is directly proportional to the charge q 1 . It turns out that this is a general fact: the potential energy W of a charge q in any electrostatic field is directly proportional to the value of q:

The value of q no longer depends on the charge, is a characteristic of the field and is called the potential:

So, the potential of a homogeneous field E at a point with the abscissa x is equal to:

Recall that the X axis coincides with the field strength line. We see that as x increases, the potential decreases. In other words, the field strength vector indicates the direction of decreasing potential. For the field potential of a point charge q at a distance r from it, we have:

The unit of measure for potential is the well-known volt. From formula (5) we see that B = J / C.

So, now we have two characteristics of the field: force (strength) and energy (potential). Each of them has its own advantages and disadvantages. Which characteristic is more convenient to use depends on the specific task.

When the charge is removed to infinity

r2 = ∞ U=U2 = 0,

charge potential energy q2,

located in the field of charge q1

on distance r

17. Potential. Potential of the field of a point charge.

Potential charge energy q in field n charges qi

Attitude U/q does not depend on the amount of charge q and is energy characteristic electrostatic field called potential.

The potential at the point of an electrostatic field is a physical quantity numerically equal to the potential energy of a single positive charge placed at this point. This is a scalar value.

in SI φ measured in Volts [V = J/C]

1 V is the potential of such a point of the field at which a charge of 1 C has an energy of 1 J.

E - [N / C = N m / C m = (J / C) (1 / m) = V / m].

Potential of the field of a point charge

The potential is a more convenient physical quantity compared to the intensity E

Potential energy of a charge in the field of a system of charges. Superposition principle for potentials.

Point charge system: q1,q2, …qn.

Distance from each charge to some point in space: r1,r2, …rn.

The work done on the charge q the electric field of the remaining charges when it moves from one point to another, is equal to the algebraic sum of the work due to each of the charges separately

ri 1 - distance from the charge qi to the initial charge position q,

ri 2 - distance from the charge qi to final charge position q.

ri 2 → ∞

Potential difference. Equipotential surfaces

When moving charge q 0+ in an electrostatic field from point 1 to point 2

r2 = ∞ → U 2 = U∞ = 0

Potential- a physical quantity determined by the work of moving a unit positive charge from a given point to infinity.

When talking about potential, they mean the potential difference ∆ φ between the considered point and the point, the potential φ which is taken as 0.

Potential φ this point has no physical meaning, since it is impossible to determine the work at this point.

Equipotential surfaces (surfaces of equal potential)

1) all points potential φ has the same meaning

2) electric field strength vector E is always normal to equipotential surfaces,

3) ∆φ between any two equipotential surfaces is the same

– For a point charge

φ = const.

r = const.

For a uniform field, the equipotential surfaces are parallel lines.

The work of moving a charge along an equipotential surface is zero.

because φ 1 = φ 2.

20. Relationship of the tension vector E and potential difference.

The work of moving a charge in an electric field:

The potential energy of the electric field depends on the coordinates x, y, z and is a function U(x,y,z).

When moving charge:

(x+dx), (y+dy), (z+dz).

Change and potential energy:

From (1)

Nabla operator (Hamilton operator).

(Brief theoretical information)

Interaction energy of point charges

The interaction energy of a system of point charges is equal to the work of external forces to create this system (see Fig. 1) by means of a slow (quasi-static) movement of charges from points infinitely distant from each other to given positions. This energy depends only on the final configuration of the system, but not on the way in which this system was created.

Based on this definition, one can obtain the following formula for the interaction energy of two point charges located in vacuum at a distance r 12 apart:

. (1)

If the system contains three fixed point charges, then the energy of their interaction is equal to the sum of the energies of all pair interactions:

where r 12 - the distance between the first and second, r 13 - between the first and third, r 23 - between the second and third charges. Similarly, the electric energy of the interaction of the system is calculated from N point charges:

For example, for a system of 4 charges, formula (2) contains 6 terms.

Electrical energy of charged conductors

The electrical energy of a solitary charged conductor is equal to the work that must be done to apply a given charge to the conductor, slowly moving it in infinitesimal portions from infinity, where initially these portions of the charge did not interact. The electrical energy of a solitary conductor can be calculated by the formula

, (3)

where q- the charge of the conductor,  - its potential. In particular, if a charged conductor has the shape of a sphere and is located in a vacuum, then its potential
and, as follows from (3), the electric energy is equal to

,

where R is the radius of the ball, q is its charge.

Similarly, the electrical energy of several charged conductors is determined - it is equal to the work of external forces to apply these charges to the conductors. For the electrical energy of the system from N charged conductors, you can get the formula:

, (4)

where and - charge and potential - conductor. Note that formulas (3), (4) are also valid in the case when the charged conductors are not in a vacuum, but in an isotropic neutral dielectric.

Using (4), we calculate the electric the energy of a charged capacitor. Denoting the charge of the positive plate q, its potential  1 , and the potential of the negative lining  2 , we get:

,

where
is the voltage across the capacitor. Given that
, the formula for the energy of a capacitor can also be represented as

, (5)

where C is the capacitance of the capacitor.

Own electric energy and interaction energy

Consider the electrical energy of two conducting balls, the radii of which R 1 , R 2 and charges q 1 , q 2. We assume that the balls are located in vacuum at a large distance compared to their radii l from each other. In this case, the distance from the center of one ball to any point on the surface of the other is approximately equal to l and the potentials of the balls can be expressed by the formulas:

,
.

We find the electrical energy of the system using (4):

.

The first term in the resulting formula is the interaction energy of the charges located on the first ball. This energy is called self-electric energy (of the first ball). Similarly, the second term is the self-electric energy of the second ball. The last term is the energy of interaction of the charges of the first ball with the charges of the second one.

At
the electric energy of interaction is significantly less than the sum of the self-energies of the balls, however, when the distance between the balls changes, the self-energies remain practically constant and the change in the total electric energy is approximately equal to the change in the interaction energy. This conclusion is valid not only for conducting balls, but also for charged bodies of arbitrary shape located on long distance from each other: the increment of the electric energy of the system is equal to the increment of the energy of interaction of the charged bodies of the system:
. Interaction energy
bodies distant from each other does not depend on their shape and is determined by formula (2).

When deriving formulas (1), (2), each of the point charges was considered as something whole and unchanged. Only the work done during the approach of such constant charges was taken into account, but not for their formation. On the contrary, when deriving formulas (3), (4), the work done when applying charges was also taken into account q i on each of the bodies of the system by transferring electricity in infinitesimal portions from infinitely distant points. Therefore, formulas (3), (4) determine the total electrical energy of the system of charges, and formulas (1), (2) only determine the electrical energy of the interaction of point charges.

Volumetric energy density of the electric field

The electrical energy of a flat capacitor can be expressed in terms of the field strength between its plates:

,

where
- the amount of space occupied by the field, S- the area of ​​the covers, d is the distance between them. It turns out that the electric energy of an arbitrary system of charged conductors and dielectrics can be expressed through tension:

, (5)

,

and the integration is carried out over the entire space occupied by the field (it is assumed that the dielectric is isotropic and
). Value w is the electrical energy per unit volume. The form of formula (5) gives reason to assume that the electrical energy is contained not in the interacting charges, but in their electric field that fills the space. Within the framework of electrostatics, this assumption cannot be verified experimentally or justified theoretically, however, consideration of alternating electric and magnetic fields makes it possible to verify the correctness of such a field interpretation of formula (5).

The forces of interaction of electric charges are conservative, therefore, the system of electric charges has potential energy.

Let two point fixed charges q 1 and q 2 be given, located at a distance r from each other. Each of the charges in the field of another charge has a potential energy

; , (4.1)

where j 1.2 and j 2.1 are, respectively, the potentials created by the charge q 2 at the location of the charge q 1 and by the charge q 1 at the location of the charge q 2.

, a . (4.3)

Consequently,

. (4.4)

In order for both charges to enter the energy equation of the system symmetrically, expression (4.4) can be written as

. (4.5)

By adding the charges q 3 , q 4 , etc. to the system of charges in series, one can make sure that in the case of N charges, the potential energy of the system

, (4.6)

where j i is the potential created at the point where q i is located by all charges, except for the i -th.

With a continuous distribution of charges in the elementary volume dV, there is a charge dq = r × dV. To determine the charge interaction energy dq, formula (4.6) can be applied, passing in it from the sum to the integral:

, (4.7)

where j is the potential at the point of the volume element dV.

It should be noted that there is a fundamental difference between formulas (4.6) and (4.7). Formula (4.6) takes into account only the energy of interaction between point charges, but does not take into account the energy of interaction of the elements of the charge of each of the point charges with each other (intrinsic energy of the point charge). Formula (4.7) takes into account both the energy of interaction between point charges and the self-energy of these charges. When calculating the interaction energy of point charges, it is reduced to volume integrals V i of point charges:

, (4.8)

where j i is the potential at any point in the volume of the i-th point charge;

j i = j i ¢ + j i с, (4.9)

where j i ¢ is the potential created by other point charges at the same point;

j i с is the potential created by parts of the i-th point charge at a given point.

Since point charges can be represented as spherically symmetrical, then

(4.10)

where W ¢ is determined by formula (4.6).

The value of the self-energy of the charges depends on the laws of charge distribution and on the magnitude of the charges. For example, with a uniform spherical charge distribution with surface density s

.

Consequently,

. (4.11)

It can be seen from formula (4.11) that for R®0 the quantity W c ®¥. This means that the self-energy of a point charge is equal to infinity. This leads to serious shortcomings of the "point charge" concept.

Thus, formula (4.6) can be used to analyze the interaction of point charges, since it does not contain their own energy. Formula (4.7) for the continuous charge distribution takes into account the entire interaction energy, and therefore is more general.

In the presence of surface charges, the form of formula (4.7) changes somewhat. The integrand of this formula is and has the meaning of the potential energy that an element of charge dq has, being at a point with potential j. This potential energy does not depend on whether dq is a volume or surface charge element. Therefore, for the surface distribution dq = s×dS. Therefore, for the field energy of surface charges

The principle of superposition.

If an electric field created by several charged bodies is investigated using a test charge, then the resulting force turns out to be equal to the geometric sum of the forces acting on the test charge from each charged body separately. Consequently, the strength of the electric field created by the system of charges at a given point in space is equal to the vector sum of the strengths of the electric fields created at the same point by the charges separately:

This property of the electric field means that the field obeys superposition principle. In accordance with Coulomb's law, the strength of the electrostatic field created by a point charge Q at a distance r from it is equal in absolute value to:

This field is called the Coulomb field. In the Coulomb field, the direction of the intensity vector depends on the sign of the charge Q: if Q is greater than 0, then the intensity vector is directed away from the charge, if Q is less than 0, then the intensity vector is directed towards the charge. The magnitude of the tension depends on the magnitude of the charge, the environment in which the charge is located, and decreases with increasing distance.

The electric field strength that a charged plane creates near its surface:

So, if in the task it is required to determine the field strength of the system of charges, then it is necessary to act according to the following algorithm:

1. Draw a picture.

2. Display the field strength of each charge separately at the desired point. Remember that tension is directed towards the negative charge and away from the positive charge.

3. Calculate each of the tensions using the appropriate formula.

4. Add up the stress vectors geometrically (i.e. vectorially).

Potential energy of interaction of charges.

Electric charges interact with each other and with an electric field. Any interaction is described by potential energy. Potential energy of interaction of two point electric charges calculated by the formula:

Pay attention to the lack of modules in the charges. For opposite charges, the interaction energy has a negative value. The same formula is also valid for the interaction energy of uniformly charged spheres and balls. As usual, in this case the distance r is measured between the centers of balls or spheres. If there are more than two charges, then the energy of their interaction should be considered as follows: divide the system of charges into all possible pairs, calculate the interaction energy of each pair and sum up all the energies for all pairs.

Problems on this topic are solved, as well as problems on the law of conservation of mechanical energy: first, the initial interaction energy is found, then the final one. If the task asks to find the work on the movement of charges, then it will be equal to the difference between the initial and final total energy of the interaction of charges. The interaction energy can also be converted into kinetic energy or into other types of energy. If the bodies are at a very large distance, then the energy of their interaction is assumed to be 0.

Please note: if the task requires finding the minimum or maximum distance between bodies (particles) during movement, then this condition will be satisfied at the moment when the particles move in the same direction at the same speed. Therefore, the solution must begin with writing the law of conservation of momentum, from which this same speed is found. And then you should write the law of conservation of energy, taking into account the kinetic energy of the particles in the second case.