Energy of interaction of electric charges. Potential energy of interaction of two charges Potential energy of interaction of a system of electric charges

If the speed v charged particle is directed at an angle a to the vector AT(Fig. 1), then its motion can be represented as a superposition: 1) uniform rectilinear motion along the field with a speed v || =v cos a; 2) uniform movement with speed v ^ =v sin a around a circle in a plane perpendicular to the field.

plane perpendicular to the field.

The radius of the circle is determined by formula (34) (in this case, it is necessary to replace v on the v ^ =v sin a). As a result of the addition of both motions, a spiral motion arises, the axis of which is parallel to the magnetic field (Fig. 1). Helix pitch

Substituting (35) into the last expression, we obtain

The direction in which the spiral twists depends on the sign of the charge of the particle.

If the speed of a charged particle makes an angle a with vector direction AT heterogeneous magnetic field, the induction of which increases in the direction of particle motion, then r and h decrease with growth B. Focusing of charged particles in a magnetic field is based on this.

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.

as 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).

14) Potential charge energy in an electric field. The work done by the forces of the electric field when moving a positive point charge q from position 1 to position 2 can be represented as a change in the potential energy of this charge:

where Wp1 and Wp2 are the potential energies of charge q in positions 1 and 2. With a small displacement of charge q in the field created by a positive point charge Q, the change in potential energy is equal to

With the final movement of the charge q from position 1 to position 2, located at distances r1 and r2 from the charge Q,

If the field is created by a system of point charges Q1, Q2,¼, Qn, then the change in the potential energy of the charge q in this field:

The above formulas allow you to find only the change in the potential energy of a point charge q, and not the potential energy itself. To determine the potential energy, it is necessary to agree at what point of the field to consider it equal to zero. For the potential energy of a point charge q located in an electric field created by another point charge Q, we obtain

where C is an arbitrary constant. Let the potential energy be equal to zero at an infinitely large distance from the charge Q (for r ® ¥), then the constant C = 0 and the previous expression takes the form

In this case, potential energy is defined as the work of moving a charge by the forces of the field from a given point to an infinitely distant one. In the case of an electric field created by a system of point charges, the potential energy of the charge q:

Potential energy of a system of point charges. 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 Qi (i = 1, 2, ... , n) in space. The interaction energy of all n charges is determined by the relation

where r i j is 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.

34. Magnetic interactions: experiments of Oersted and Ampère; a magnetic field; Lorentz force, magnetic field induction; magnetic field lines; magnetic field created by a point charge moving at a constant speed.

A magnetic field- a force field acting on moving electric charges and on bodies with a magnetic moment, regardless of the state of their movement, the magnetic component of the electromagnetic field

The magnetic field can be created by the current of charged particles and/or by the magnetic moments of electrons in atoms (and by the magnetic moments of other particles, although to a much lesser extent) (permanent magnets).

Oersted's experience showed that electric currents could act on magnets, but the nature of the magnet at that time was completely mysterious. Ampere and others soon discovered the interaction of electric currents with each other, manifesting itself, in particular, as an attraction between two parallel wires, through which currents flow in the same direction. This led Ampere to the hypothesis that there are constantly circulating electric currents in magnetic matter. If such a hypothesis is correct, then the result of Oersted's experiment can be explained by the interaction of the galvanic current in the wire with microscopic currents that impart special properties to the compass needle.

Lorentz force- the force with which, in the framework of classical physics, the electromagnetic field acts on a point charged particle. Sometimes the Lorentz force is called the force acting on a charge moving at a speed only from the side of the magnetic field, often the full force - from the side of the electromagnetic field in general, in other words, from the side of the electric and magnetic fields. Expressed in WSI as:

For a continuous charge distribution, the Lorentz force takes the form:

where dF- force acting on a small element dq.

INDUCTION OF A MAGNETIC FIELD is a vector quantity, which is a force characteristic of a magnetic field (its action on charged particles) at a given point in space. Determines the force with which the magnetic field acts on a charge moving at speed.

More specifically, is such a vector that the Lorentz force acting from the side of the magnetic field on a charge moving at speed is equal to

where the oblique cross denotes the vector product, α is the angle between the velocity and magnetic induction vectors (the direction of the vector is perpendicular to both of them and is directed to the correct gimlet).

36. The effect of magnetic fields on electric currents: the Biot-Savart-Laplace-Ampère law and its application to calculate the force acting from a uniform magnetic field on a segment of a thin straight current-carrying conductor; Ampère formula and its significance in metrology.

Consider an arbitrary conductor in which currents flow:

dF=* ndV=* dV

Z-n Bio-Savart-Ampere for bulk current: dF=jBdVsin. dF perpendicular ,those. directed towards us. Let's take a thin conductor: , then for a linear electric current, the s-n will be written in the form: dF= I, i.e.dF= IBdlsin.

Task 1! There is a uniform magnetic field. In it, nah-I have a piece of wire that has l and I.

d= I , dF= IBdlsin, F= IBsin= IBlsin- Ampere power.

1 Ampere-strength of current, during the flow of which 2 || long, thin conductors located at a distance of 1 m from each other, a force equal to 2 * 10 ^ -7 N acts on each meter of their length.

Task 2! There are 2 || long conductors, where l >> d,thend=, dd, . Then the Ampere function: *l.

37. Magnetic dipole: physical model and magnetic moment of the dipole; magnetic field created by a magnetic dipole; forces acting from homogeneous and inhomogeneous magnetic fields on a magnetic dipole.

DIPOLE MAGNETIC an analogue of an electric dipole, which can be thought of as two point magnets. charge located at a distance l from each other. It is characterized by a dipole moment equal in magnitude and directed from.

The fields created by equal D. m. outside the region of sources in vacuum (or in any other medium, the magnetic permeability of which = 1) are the same, but in media with coincidence is achieved if only we accept that, i.e., assume that the dipole moment of the charge D. m. depends on the permeability

38. Gauss's theorem for a magnetic field: integral and differential forms, the physical meaning of the theorem. Relativistic nature of the magnetic field: magnetic interactions as a relativistic consequence of electrical interactions; mutual transformations of electric and magnetic fields.

The absence of magnetic charges in nature leads to the fact that the lines of the vector AT have no beginning or end. Vector flow AT through a closed surface must be equal to zero. Thus, for any magnetic field and an arbitrary closed surface S condition

This formula expresses the Gauss theorem for the vector AT : the flux of the magnetic induction vector through any closed surface is zero.

In integral form

1. The flow of the electric displacement vector through any closed surface surrounding a certain volume is equal to the algebraic sum of free charges located inside this surface

A vector is a characteristic of a field that does not depend on the dielectric properties of the medium.

In differential form

Let the volume contain

where is the volume average density. Then

When contracting the volume to a point

- Gauss theorem in differential form

39. The theorem on the circulation of the vector of magnetic induction of a stationary magnetic field for vacuum: integral and differential forms, the physical meaning of the theorem; application of the theorem for the calculation of magnetic fields on the example of a magnetic field created by an infinitely long solenoid with current.

Theorem. Circulation of the magnetic induction vector B in a closed loopLequal to the algebraic sum of the currents covered by this circuitLmultiplied by μ 0 .

Examples:

I 3

I 1 I 2

- current outside the circuit.

Applying the principle of superposition to magnetic fields, we get:

If the currents flow in a continuous medium, we get:

Stokes' theorem: where S - surface bounded by a contour L .

- theorem on the circulation of the magnetic induction vector.

for electrostatic field

electrostatic field - potential, there are sources of the field - charges.

2) for magnetic field

the magnetic field is not potential, but vortex, there are no magnetic charges.

Solenoid - a coil with turns tightly wound to each other on a cylindrical core, whilel>> D(if the solenoid is considered infinite).

- magnetic field induction

toroid, wheren- the number of turns per unit length of the center line

40. Magnetics. Magnetization of matter: the physical essence of the phenomenon; Ampère's hypothesis about molecular currents; magnetization currents, magnetization (magnetization vector); connection of the magnetization vector with surface and bulk magnetization currents.

Magnetics Substances that can become magnetized when placed in an external electric field. Atoms have magnetic moments. In the absence of an external magnetic field, the magnetic moments of the atoms are randomly oriented and the total magnetic moment of the substance is zero. When making a substance in ext. magn. field, magnetic the moments of the atoms are oriented predominantly in one direction, as a result of which the total moment is nonzero and the substance is magnetized. The degree of magnetization of magnets is characterized by the value:

Magnetization of a magnet (magnetization vector)

The magnetized substance creates its own magnetic field with induction B 0, then the induction of the resulting magnetic field

Magnetization of a magnet

B 0 cylindrical shape

Magnetic field strength

x<0, μ<1 – диамагнетики

x>0, μ>1 – paramagnets

x>>0, μ>>1 – ferromagnets

Diamagnets - substances whose magnetic moments of atoms, in the absence of an external magnetic field, are equal to zero (non-ferrous gases, glass, water, gold, silver, copper, mercury). For diamagnets, the magnetic susceptibility does not depend on temperature.

Paramagnets - substances whose magnetic moments of atoms are different from zero (oxygen, nitric oxide, aluminum, platinum)

Ampere suggested that certain currents circulate inside the substance, which he called molecular- these are the currents associated with the orbital motion of electrons.

THEN. each electron that moves in the orbit of an atom creates its own current.

The action of a magnetic field on a current-carrying conductor. Z-n Ampera.

Let us show that Ampere's law follows from the Lorentz force. Each charged particle is affected by the Lorentz force.

Calculate the force acting on the element

Force per current element

Force acting

on the conductor element with

current, the power of Ampere.

45 Electromagnetic induction: Faraday's experiments on electromagnetic induction; the physical essence of the phenomenon; Faraday's law of electromagnetic induction and its physical justification, Lenz's rule; operating principle of the fluxmeter.

Discovered by Faraday in 1831 electromagnetic induction called the phenomenon of the occurrence of current in a closed conducting circuit when the magnetic flux penetrating this circuit changes.

EMF of electromagnetic induction.

Lenz's rule: the induction current has such a direction that its magnetic field opposes the change in the magnetic flux that causes this current.

- s-n of electromagnetic induction (s-n of Faraday).

Toki Foucault- eddy currents that occur in a conductive medium when the magnetic flux penetrating this medium changes.

The magnitude of the Foucault currents depends on the frequency

magnetic flux changes and

material resistance. Eddy currents

Foucault heat up a massive conductor.

Flux linkage. Loop inductance. solenoid inductance.

N B Let there be a solenoid.

(magnetic flux associated

I with one turn).

– flux linkage, the magnetic flux associated with all turns. It has been experimentally established that the flux linkage is proportional to the current:

– inductance

is the induction of the magnetic field of the solenoid.

is the inductance of the solenoid, where

(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).

Potential interaction energy of a system of point charges and total electrostatic energy of a system of charges

Animation

Description

The potential energy of interaction of two point charges q 1 and q 2 located in vacuum at a distance r 12 from each other can be calculated from:

(1)

Consider a system consisting of N point charges: q 1 , q 2 ,..., q n .

The interaction energy of such a system is equal to the sum of the interaction energies of charges taken in pairs:

. (2)

In formula 2, the summation is performed over the indices i and k (i № k ). Both indices run, independently of each other, from 0 to N . Terms for which the value of index i coincides with the value of index k are not taken into account. The coefficient 1/2 is set because the potential energy of each pair of charges is taken into account twice during summation. Formula (2) can be represented as:

, (3)

where j i is the potential at the location of the i-th charge, created by all other charges:

.

The interaction energy of a system of point charges, calculated by formula (3), can be both positive and negative. For example, it is negative for two point charges of opposite sign.

Formula (3) does not determine the total electrostatic energy of a system of point charges, but only their mutual potential energy. Each charge q i , taken separately, has electrical energy. It is called the self-energy of the charge and represents the energy of mutual repulsion of infinitely small parts into which it can be mentally divided. This energy is not taken into account in formula (3). Only the work expended on the convergence of charges q i is taken into account, but not on their formation.

The total electrostatic energy of a system of point charges also takes into account the work done to form charges q i from infinitesimal portions of electricity transferred from infinity. The total electrostatic energy of a system of charges is always positive. This is easy to show by the example of a charged conductor. Considering a charged conductor as a system of point charges and taking into account the same potential value at any point of the conductor, from formula (3) we obtain:

This formula gives the total energy of a charged conductor, which is always positive (when q>0, j>0, therefore W>0, if q<0 , то j <0 , но W>0 ).

Timing

Initiation time (log to -10 to 3);

Lifetime (log tc -10 to 15);

Degradation time (log td -10 to 3);

Optimal development time (log tk -7 to 2).

Diagram:

Technical realizations of the effect

Technical implementation of the effect

To observe the interaction energy of a system of charges, it is sufficient to hang two light conductive balls on strings at a distance of about 5 cm from each other and charge them from a comb. They will deviate, that is, they will increase their potential energy in the gravitational field of the earth, which is done due to the energy of their electrostatic interaction.

Applying an effect

The effect is so fundamental that, without exaggeration, we can assume that it is applied to any electrical and electronic equipment that uses charge storage devices, that is, capacitors.

Literature

1. Saveliev I.V. Course of General Physics.- M.: Nauka, 1988.- V.2.- S.24-25.

2. Sivukhin D.V. General course of physics.- M.: Nauka, 1977.- V.3. Electricity.- S.117-118.

Keywords

• electric charge
• point charge
• potential
• potential interaction energy
• total electrical energy

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