Potential energy of the conductor. Energy of a charged conductor and capacitor

The energy of a charged conductor is numerically equal to the work that external forces must do to charge it W=A. When the charge dq is transferred from infinity to the conductor, work dA is performed against the forces of the electrostatic field (to overcome the Coulomb repulsive forces between like charges): dA=jdq=Cjdj.

To charge a body from zero potential to potential j, work is required. The energy of a charged conductor is equal to the work that must be done to charge it: .

The expression is called the self-energy of a charged conductor.

An increase in the potential j of the conductor during its charging is accompanied by an increase in the electrostatic field, and the field strength increases. It is natural to assume that the self-energy of a charged conductor is the energy of its electrostatic field. Let's check this assumption on the example of a uniform field of a flat capacitor. By repeating the course of the above calculation, it is easy to obtain the energy of a charged flat capacitor ,

where Dj is the potential difference of its plates. Let us substitute into this formula the expressions for the capacitance of a flat capacitor and the potential difference between the plates. Then for the energy we get , where V=Sd is the volume of the electrostatic field between the capacitor plates.

It follows that the self-energy of a charged flat capacitor is proportional to V to the volume of its field and strength. Therefore, it is necessary to consider that the electrostatic field has energy. The volumetric energy density of the electric field or the energy per unit volume is , . Where is the energy of the electrostatic field localized and what is its carrier - charges or the field itself? The answer to this question can only be given by experience. However, electrostatics cannot answer this question, because it studies fields of fixed charges that are constant in time, i.e. In electrostatics, fields and charges are inseparable from each other.

Experiments have shown that time-varying electric fields can exist separately, regardless of the charges that excite them. They propagate in space in the form of waves capable of carrying energy. It follows that the energy is localized in the field and the field is the carrier of electrical energy.

1. Energy of a system of fixed point charges. Electrostatic interaction forces are conservative (see § 57); therefore, the system of charges has potential energy. Find the potential energy of a system of two fixed point charges Q 1 and Q 2 , located at a distance r from each other. Each of these charges in the field of the other has a potential energy (see 58.2) and (58.5)):

Where j 12 and j 21 - respectively, the potentials created by the charge Q 2 at the location of the charge Q 1 and charge Q 1 at the location of the charge Q 2 . According to (58.5),

That's why W 1 = W 2 = W And

By adding to the system of two charges in series charges Q 3 , Q 4 , ... , it can be verified that in the case n of fixed charges, the interaction energy of a system of point charges is equal to

(69.1)

Where j i - potential created at the point where the charge is located Q i , all charges except i th.

2. Energy of a charged solitary conductor. Let there be a solitary conductor, the charge, capacitance and potential of which are respectively equal Q, C, j. Let's increase the charge of this conductor by d Q. To do this, it is necessary to transfer the charge d Q from infinity to a solitary conductor, having spent on this work equal to

To charge a body from zero potential to j, work needs to be done

(69.2)

The energy of a charged conductor is equal to the work that must be done to charge this conductor:

Formula (69.3) can also be obtained from the fact that the potential of the conductor at all its points is the same, since the surface of the conductor is equipotential. Assuming the potential of the conductor is equal j, from (69.1) we find

Where - conductor charge.

3. Energy of a charged capacitor. Like any charged conductor, a capacitor has an energy that, in accordance with formula (69.3), is equal to

Where Q- capacitor charge, WITH - its capacity DJ- potential difference between the capacitor plates.

Using expression (69.4), one can find mechanical (ponderomotive) the force with which the plates of a capacitor attract each other. For this, we assume that the distance X between the plates varies, for example, by the value d x. Then the acting force does work d A=F d x due to a decrease in the potential energy of the system F d x=- d W, where

(69.5)

Substituting expression (69.3) into (69.4), we obtain

(69.6)

Differentiating at a specific energy value (see (69.5) and (69.6)), we find the required force:

where the minus sign indicates that the force F is the force of attraction.

4. Energy of the electrostatic field. We transform formula (69.4), which expresses the energy of a flat capacitor in terms of charges and potentials, using the expression for the capacitance of a flat capacitor ( C=e 0 eS/d) and the potential difference between its plates (D j=Ed. Then

(69.7)

Where V=Sd- volume of the condenser. Formula (69.7) shows that the energy of a capacitor is expressed in terms of a quantity characterizing the electrostatic field, - tension E.

Bulk density energy of the electrostatic field (energy per unit volume)

(69.8)

Expression (69.8) is valid only for isotropic dielectric, for which relation (62.2) holds: P =æ e 0 E.

Formulas (69.4) and (69.7), respectively, relate the energy of the capacitor with charge on its covers and with field strength. Naturally, the question arises about the localization of electrostatic energy and what is its carrier - charges or field? The answer to this question can only be given by experience. Electrostatics studies the fields of fixed charges that are constant in time, i.e., in it the fields and the charges that caused them are inseparable from each other. Therefore, electrostatics cannot answer the questions posed. Further development of theory and experiment showed that time-varying electric and magnetic fields can exist separately, regardless of the charges that excited them, and propagate in space in the form of electromagnetic waves, able transfer energy. This convincingly confirms the main position short-range theory that energy is localized in a field So what carrier energy is field.

Chapter 10

§ 70. Electric current, strength and current density

IN electrodynamics- a section of the doctrine of electricity, which deals with phenomena and processes caused by the movement of electric charges or macroscopic charged bodies - the most important concept is the concept of electric current. electric shock Any ordered (directed) movement of electric charges is called. In a conductor under the action of an applied electric field E free electric charges move: positive - along the field, negative - against the field (Fig. 146, A), i.e., an electric current arises in the conductor, called conduction current. If the ordered movement of electric charges is carried out by moving a charged macroscopic body in space (Fig. 146, b) then the so-called convection current.

For the emergence and existence of an electric current, it is necessary, on the one hand, the presence of free current carriers- charged particles capable of moving in an orderly manner, and on the other hand - the presence of an electric field, whose energy, somehow replenished, would be spent on their orderly movement. For the direction of current conditionally take the direction of travel positive charges.

The quantitative measure of electric current is current strength I scalar physical quantity determined by the electric charge passing through the cross section of the conductor per unit time:

If the strength of the current and its direction do not change with time, then such a current is called permanent. For DC

Where Q- electric charge passing over time t through the cross section of the conductor. The unit of current is the ampere (A).

The physical quantity determined by the strength of the current passing through the unit area of ​​the cross section of the conductor, perpendicular to the direction of the current, is called current density:

We express the current strength and density in terms of the speed á v ñ ordered motion of charges in a conductor. If the concentration of current carriers is n and each carrier has an elementary charge e(which is not necessary for ions), then in time dt through the cross section S conductor charge is transferred dQ=ne avy S d t. Current strength

and current density

(70.1)

Current density - vector, oriented in the direction of the current, i.e. the direction of the vector j coincides with the direction of the ordered movement of positive charges. The unit of current density is ampere per square meter (A / m 2).

Current through an arbitrary surface S defined as the vector flow j, i.e.

(70.2)

where d S=n d S (n- unit normal vector to the area d S, component with vector j angle a).

§ 71. External forces. Electromotive force and voltage

If only the forces of the electrostatic field act on the current carriers in the circuit, then the carriers move (they are assumed to be positive) from points with a higher potential to points with a lower potential. This will lead to equalization of potentials at all points in the circuit and to the disappearance of the electric field. Therefore, for the existence of direct current, it is necessary to have a device in the circuit that can create and maintain a potential difference due to the work of forces of non-electrostatic origin. Such devices are called current sources. Forces non-electrostatic origin, acting on charges from current sources are called third-party.

The nature of outside forces can be different. For example, in galvanic cells they arise due to the energy of chemical reactions between electrodes and electrolytes; in the generator - due to the mechanical energy of rotation of the generator rotor, etc. The role of the current source in the electrical circuit, figuratively speaking, is the same as the role of the pump, which is necessary for pumping fluid in the hydraulic system. Under the action of the generated field of external forces, electric charges move inside the current source against the forces of the electrostatic field, due to which a potential difference is maintained at the ends of the circuit and a constant electric current flows in the circuit.

External forces do work to move electric charges. The physical quantity determined by the work done by external forces when moving a unit positive charge is called electromotive force (emf), operating in the circuit:

(71.1)

This work is done at the expense of the energy expended in the current source, so the value can also be called the electromotive force of the current source included in the circuit. Often, instead of saying: “external forces act in the circuit”, they say: “emf acts in the circuit”, i.e. the term “electromotive force” is used as a characteristic of external forces. The emf, like the potential, is expressed in volts (cf. (84.9) and (97.1)).

third party force F st acting on the charge Q 0 , can be expressed as

Where Eating- field strength of external forces. The work of external forces to move the charge Q 0 in a closed section of the circuit is equal to

(71.2)

Dividing (71.2) by Q 0 , we get an expression for e. d.s. acting in the chain:

i.e., the emf acting in a closed circuit can be defined as the circulation of the field strength vector of external forces. E.m.f. acting on the site 1 -2 , is equal to

(71.3)

per charge Q 0 in addition to external forces, the forces of the electrostatic field also act F e= Q 0 E. Thus, the resulting force acting in the circuit on the charge Q 0 is equal to

The work done by the resultant force on the charge Q 0 on site 1 -2 , is equal to

Using expressions (97.3) and (84.8), we can write

(71.4)

For a closed circuit, the work of electrostatic forces is zero (see § 57), therefore, in this case

voltage U Location on 1 -2 called a physical quantity determined by the work done by the total field of electrostatic (Coulomb) and external forces when moving a single positive charge in a given section of the circuit. Thus, according to (71.4),

The concept of voltage is a generalization of the concept of potential difference: the voltage at the ends of a circuit section is equal to the potential difference in the event that E.m.f. does not act on this section, i.e., there are no external forces.

§ 72. Ohm's law. Conductor resistance

The German physicist G. Ohm (1787; -1854) experimentally established that the current strength I, flowing through a homogeneous metal conductor (i.e., a conductor in which no external forces act), is proportional to the voltage U at the ends of the conductor:

(72.1)

Where R- electrical resistance of the conductor.

Equation (72.1) expresses Ohm's law for a circuit section(not containing a current source): the amount of current in a conductor is directly proportional to the applied voltage and inversely proportional to the resistance of the conductor. Formula (72.1) allows you to set the unit of resistance - ohm(Ohm): 1 Ohm - the resistance of such a conductor in which a direct current of 1 A flows at a voltage of 1 V.

Value

called electrical conductivity conductor. Conductivity unit - Siemens(Sm): 1 Sm - conductivity of a section of an electrical circuit with a resistance of 1 ohm.

The resistance of conductors depends on its size and shape, as well as on the material from which the conductor is made. For a homogeneous linear conductor, the resistance R directly proportional to its length l and inversely proportional to its cross-sectional area S:

(72.2)

Where r- coefficient of proportionality characterizing the material of the conductor and called specific electrical resistance. The unit of electrical resistivity is ohm×meter (Ohm×m). Silver (1.6 × 10 -8 Ω × m) and copper (1.7 × 10 –8 Ω × m) have the lowest resistivity. In practice, along with copper, aluminum wires are used. Although aluminum has a higher resistivity than copper (2.6 × 10 -8 Ohm × m), it has a lower density than copper.

Ohm's law can be represented in differential form. Substituting the expression for resistance (72.2) into Ohm's law (72.1), we obtain

(72.3)

where is the reciprocal of the resistivity,

called electrical conductivity conductor materials. Its unit is siemens per meter (S/m).

Given that U/l= E - electric field strength in the conductor, I/S = j - current density, formula (72.3) can be written as

(72.4)

Since in an isotropic conductor the current carriers at each point move in the direction of the vector E, then the directions j And E match up. Therefore, formula (98.4) can be written in vector form

Expression (72.5) - Ohm's law in differential form, relating the current density at any point inside the conductor with the electric field strength at the same point. This relation is also valid for variable fields.

Experience shows that, as a first approximation, the change in resistivity, and hence resistance, with temperature is described by a linear law:

Where r And r 0 , R And R 0 - respectively, the resistivity and resistance of the conductor at t and 0°С, a -temperature coefficient of resistance, for pure metals (at not very low temperatures) close to 1/273 K–1. Therefore, the temperature dependence of the resistance can be represented as

Where T - thermodynamic temperature.

The qualitative course of the temperature dependence of the resistance of the metal is shown in Fig. 147 (curve 1 ). Subsequently, it was found that the resistance of many metals (for example, Al, Pb, Zn, etc.) and their alloys at very low temperatures T K(0.14-20 K), called critical, characteristic of each substance decreases abruptly to zero (curve 2 ), i.e., the metal becomes an absolute conductor. For the first time this phenomenon, called superconductivity, was discovered in 1911 by G. Kamerling-Onnes for mercury. The phenomenon of superconductivity is explained on the basis of quantum theory. The practical use of superconducting materials (in the windings of superconducting magnets, in computer memory systems, etc.) is difficult because of their low critical temperatures. Currently, ceramic materials with superconductivity at temperatures above 100 K have been discovered and are being actively studied.

The action is based on the dependence of the electrical resistance of metals on temperature resistance thermometers, which allow measuring the temperature with an accuracy of 0.003 K by a graduated relationship of resistance to temperature. thermistors. They allow you to measure temperatures with an accuracy of millionths of a kelvin.

§ 73. Work and current power. Joule-Lenz law

Consider a homogeneous conductor, to the ends of which a voltage is applied U. For "time d t charge d is transferred through the cross section of the conductor q=I d t. Since the current is the movement of charge d q under the action of an electric field, then, according to the formula (84.6), the work of the current

(73.1)

If the conductor resistance R, then, using law (72.1), we obtain

(73.2)

From (73.1) and (73.2) it follows that the current power

(73.3)

If current is expressed in amperes, voltage is in volts, resistance is in ohms, then the work of the current is expressed in joules, and the power is in watts. In practice, off-system units of current work are also used: watt-hour (Wh) and kilowatt-hour (kWh). 1 W×h - operation of a current with a power of 1 W for 1 hour; 1 W×h=3600 W×s=3.6×10 3 J; 1 kWh=10 3 Wh= 3.6×10 6 J.

If the current passes through motionless metal conductor, then all the work of the current goes to heat it and, according to the law of conservation of energy,

(73.4)

Thus, using expressions (73.4), (73.1) and (73.2), we obtain

(73.5)

Expression (73.5) is Joule's law-Lenz, experimentally established independently by J. Joule and E. X. Lenz.

Let us single out in the conductor an elementary cylindrical volume d V= d S d l(the axis of the cylinder coincides with the direction of the current), the resistance of which According to the Joule-Lenz law, in time d t heat will be released in this volume

The amount of heat released per unit time per unit volume is called specific thermal current power. She is equal

(73.6)

Using the differential form of Ohm's law ( j=gЕ) and ratio r= 1/g, we get

(73.7)

Formulas (73.6) and (73.7) are a generalized expression Joule-Lenz law in differential form, suitable for any conductor.

The thermal effect of the current is widely used in technology, which began with the discovery in 1873 by the Russian engineer A. N. Lodygin (1847-1923) of an incandescent lamp. The action of electric muffle furnaces, an electric arc (discovered by the Russian engineer V.V. Petrov (1761-1834)), contact electric welding, household electric heaters, etc. is based on heating conductors with electric current.

§ 74. Ohm's law for an inhomogeneous section of a chain

We considered Ohm's law (see (98.1)) for a homogeneous section of the circuit, i.e., one in which the emf does not exist. (no third party forces). Now consider heterogeneous section of the chain, where the current e.m.f. Location on 1 -2 denote by a the potential difference applied at the ends of the section - through j 1 -j 2 .

If the current passes through motionless conductors forming a section 1-2, then work A 12 of all forces (third-party and electrostatic) performed on current carriers, according to the law of conservation and transformation of energy, is equal to the heat released in the area. The work of forces performed when moving a charge Q 0 on site 1 -2 , according to (71.4),

emf like the current I, - the value is scalar. It must be taken either with a positive or with a negative sign, depending on the sign of the work done by outside forces. If emf contributes to the movement of positive charges in the chosen direction (in the direction 1-2 ), then > 0. If the emf prevents positive charges from moving in that direction.< 0.

During t heat is released in the conductor (see (73.5))

From formulas (74.1) and (74.2) we obtain

(74.4)

Expression (74.3) or (74.4) is Ohm's law for an inhomogeneous section of a circuit in integral form, which is generalized Ohm's law.

If in this section of the chain no current source(=0), then from (74.4) we arrive at Ohm's law for a homogeneous section of the chain (72.1):

(in the absence of external forces, the voltage at the ends of the section is equal to the potential difference (see § 71)).

If the electrical circuit closed then the chosen points 1 And 2 match, j 1 =j 2; then from (74.4) we get Ohm's law for a closed circuit:

where is the emf acting in the circuit, R- the total resistance of the entire circuit. In general R=r+R 1 , Where r- internal resistance of the current source, R 1 - external circuit resistance. Therefore, the law for a closed circuit will have the form

If the chain open and, therefore, there is no current in it ( I= 0), then from Ohm's law (74.4) we obtain that =j 1 -j 2 , i.e., the emf acting in an open circuit is equal to the potential difference at its ends. Therefore, in order to find the emf. current source, it is necessary to measure the potential difference at its terminals with an open circuit.

The charge q located on some conductor can be considered as a system of point charges q. Earlier we obtained (3.7.1) an expression for the interaction energy of a system of point charges:

The conductor surface is equipotential. Therefore, the potentials of those points where the point charges q i are located are the same and equal to the potential j of the conductor. Using formula (3.7.10), we obtain the expression for the energy of a charged conductor:

. (3.7.11)

Any of the following formulas (3.7.12) gives the energy of a charged conductor:

. (3.7.12)

So, it is logical to raise the question: where is the energy localized, what is the carrier of energy - charges or field? Within the limits of electrostatics, which studies fields of stationary charges that are constant in time, it is impossible to give an answer. Constant fields and the charges that caused them cannot exist separately from each other. However, time-varying fields can exist independently of the charges that excite them and propagate in the form of electromagnetic waves. Experience shows that electromagnetic waves carry energy. These facts force us to admit that the energy carrier is the field.

Literature:

Main 2, 7, 8.

Add. 22.

Control questions:

1. Under what conditions can the interaction forces of two charged bodies be found according to the Coulomb law?

2. What is the flow of the electrostatic field strength in vacuum through a closed surface?

3. Which electrostatic fields can be conveniently calculated based on the Ostrogradsky-Gauss theorem?

4. What can be said about the strength and potential of the electrostatic field inside and near the surface of the conductor?

1.4.2. The energy of a charged conductor

When charging a certain conductor, it is necessary to do some work against the Coulomb repulsive forces between like electric charges. This work goes to increase the electrical energy of a charged conductor, which in this case is similar to potential energy in mechanics.

Consider a conductor with electric capacitance , charge and potential . Work done against the forces of an electrostatic field when transferring a charge
from infinity to the conductor is

.

To charge a body from zero potential to potential , work needs to be done
. It is clear that the energy of a charged body is equal to the work that needs to be done to charge this body:
.

energy called the self-energy of a charged body. It is clear that self-energy is nothing but the energy of the electrostatic field of this body.

1.4.3. The energy of a charged capacitor. Volumetric energy density of the electrostatic field

Let the potential of the capacitor plate on which the charge is
, is equal to , and the potential of the plate on which the charge is located
,. The energy of such a system of charges, that is, is equal to the self-energy of the system of charges, where is the voltage between the capacitor plates,
.

Consider a flat capacitor. The energy contained in a unit volume of an electrostatic field is called the volume energy plane. This volume plane must be the same at all points of a homogeneous field, and the total energy of the field is proportional to its volume. It is known that
,
, then for the energy we have:
, But
- the volume of the electrostatic field between the capacitor plates, that is
. Then the volume energy density uniform electrostatic field of the capacitor is equal to
, and is determined by its tension or displacement. In the case of inhomogeneous electric fields

Find the energy of a spherical capacitor. On distance from the center of the charged ball, the strength of its electrostatic field is equal to
. Consider an infinitely thin spherical layer enclosed between spheres of radii And
. The volume of such a layer:
. Layer energy
hence,

.

Then the total energy of the charged ball is:

,

Where is the radius of the ball. Ball capacity
, hence,
- the energy of the electrostatic field of a spherical capacitor is equal to its own energy, since a charged body therefore possesses electrical energy, because when it was charged, work was done against the forces of the electrostatic field created by it.

1.4.4. Energy of a polarized dielectric. Volumetric energy density of the electric field in a dielectric

Consider a homogeneous isotropic dielectric in an external electric field. The process of polarization is associated with the work on the deformation of electron orbits in atoms and molecules and on the rotation of the axes of dipole molecules along the field. It is clear that a polarized dielectric must have a store of electrical energy.

If the field strength created in a vacuum
, then the volumetric energy density of this field at a point with intensity is equal to:

Let us prove that the volume energy density of a polarized dielectric at this point is expressed by the formula:
.

Consider a dielectric with nonpolar molecules. The molecules of such a dielectric are elastic dipoles. The electric moment of an elastic dipole in a field with intensity , is equal to
, Where is the dipole polarizability, or in scalar form:

, (1.4.1)

Where
are the charge and arm of the dipole.

per charge force acting from the field
, which, as the dipole length increases by
does the job
. From expression (1.4.1) we get:
, That's why

. (1.4.2)

To find a job field during the deformation of one elastic dipole, it is necessary to integrate the expression (1.4.2):

.

Job is equal to the potential energy possessed by an elastic dipole in an electric field with strength . Let is the number of dipoles per unit volume of the dielectric. Then the potential energy of all these dipoles, that is, the volume energy density of a polarized dielectric, is equal to:
. However
is the modulus of the polarization vector, then
. It is known that
, And
, Then
, which was to be proved.

1. Energy of a system of fixed point charges. The electrostatic interaction forces are conservative; therefore, the system of charges has potential energy. Let's find the potential energy of a system of two point charges Q 1 and Q 2 located at a distance r from each other. Each of these charges in the field of the other has a potential energy:

where φ 12 and φ 21 are, respectively, the potentials created by the charge Q 2 in charge point Q1 and charge Q1 at the location of the charge Q2. The field potential of a point charge is:

By adding to the system of two charges in series charges Q 3 , Q 4 , …, one can make sure that in the case of n stationary charges, the interaction energy of a system of point charges is equal to

(3)

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

2. The energy of a charged solitary conductor. Let there be a solitary conductor, the charge, capacitance and potential of which are respectively equal Q, C, φ. Let's increase the charge of this conductor by dQ. To do this, it is necessary to transfer the charge dQ from infinity to a solitary conductor, spending on this work equal to

To charge the body from zero potential to j, it is necessary to do work

The energy of a charged conductor is equal to the work that must be done to charge this conductor:

(4)

This formula can also be obtained from the fact that the potential of the conductor at all its points is the same, since the surface of the conductor is equipotential. Assuming the potential of the conductor to be equal to j, from (3) we find

where is the charge of the conductor.

3. Energy of a charged capacitor. Like any charged conductor, a capacitor has an energy that, in accordance with formula (4), is equal to

(5)

Where Q- capacitor charge, WITH- its capacitance, Dj - potential difference between the plates.

Using expression (5), one can find mechanical force from which the plates of a capacitor attract each other. For this, we assume that the distance X between the plates varies, for example, by the value dx. Then the force does work

due to a decrease in the potential energy of the system

F dx = -dW,

(6)

Substituting in (5) into the capacitance formula of a flat capacitor, we obtain

(7)

Differentiating at a specific energy value (see (6) and (7)), we find the desired force:

,

where the minus sign indicates that the force F is the force of attraction.

4. Electrostatic field energy.

Let us transform formula (5), which expresses the energy of a flat capacitor in terms of charges and potentials, using the expression for the capacitance of a flat capacitor (C = e 0 eS/d) and the potential difference between its plates (Dj = Ed). Then we get

(8)

Where V=Sd is the volume of the condenser. This formula shows that the energy of a capacitor is expressed in terms of a quantity characterizing the electrostatic field, - tension E.

Bulk density energy of the electrostatic field (energy per unit volume)

This expression is only valid for isotropic dielectric, for which the relation is fulfilled: Р = ce 0 E.

Formulas (5) and (8) respectively relate the energy of the capacitor with charge on its covers and with field strength. Naturally, the question arises about the localization of electrostatic energy and what is its carrier - charges or fields? The answer to this question can only be given by experience. Electrostatics studies the fields of fixed charges that are constant in time, i.e., in it the fields and the charges that caused them are inseparable from each other. Therefore, electrostatics cannot answer the questions posed. Further development of theory and experiment showed that time-varying electric and magnetic fields can exist separately, regardless of the charges that excited them, and propagate in space in the form of electromagnetic waves, able transfer energy. This convincingly confirms the main position the theory of short-range action on the localization of energy in a field So what carrier energy is field.

Electric dipoles

Two equal charges of opposite sign, + Q And- Q, located at a distance l from each other, form electric dipole. Value Ql called dipole moment and is denoted by the symbol R. Many molecules have a dipole moment, for example, the diatomic CO molecule (the C atom has a small positive charge, and O has a small negative charge); despite the fact that the molecule as a whole is neutral, charge separation occurs in it due to the unequal distribution of electrons between the two atoms. (Symmetrical diatomic molecules, such as O 2 , do not have a dipole moment.)

Consider first a dipole with moment ρ = Ql, placed in a uniform electric field of strength Ε. The dipole moment can be represented as a vector p, equal in absolute value to Ql and directed from a negative charge to a positive one. If the field is uniform, then the forces acting on the positive charge QE, and negative, QE, do not create a net force acting on the dipole. However, they give rise to torque, whose value relative to the middle of the dipole ABOUT is equal to

or in vector notation

As a result, the dipole tends to rotate so that the vector p is parallel to E. Work W, performed by an electric field over a dipole when the angle θ changes from q 1 to q 2 is given by

As a result of the work done by the electric field, the potential energy decreases U dipole; if put U= 0 when p^Ε (θ = 90 0), then

U=-W=-pEcosθ = -p Ε.

If the electric field heterogeneous, then the forces acting on the positive and negative charges of the dipole may turn out to be unequal in magnitude, and then, in addition to the torque, the resulting force will also act on the dipole.

So, we see what happens to an electric dipole placed in an external electric field. Let us now turn to the other side of the matter.

rice. Electric field created by an electric dipole.

Assume that there is no external field and determine the electric field created by the dipole itself(capable of acting on other charges). For simplicity, we confine ourselves to points located on the perpendicular to the middle of the dipole, like the point Ρ in fig. ???, located at a distance r from the middle of the dipole. (Note that r in Fig.??? is not the distance from each of the charges to R, which is equal to (r 2 +/ 2 /4) 1/2) Electric field strength at: point Ρ is equal to

Ε = Ε + + Ε - ,

where E + and E - are the field strengths created respectively by positive and negative charges, equal to each other in absolute value:

Their y-components at a point Ρ cancel each other out, and the absolute value of the electric field strength Ε is equal to

,

[along the perpendicular to the middle of the dipole].

away from the dipole (r»/) this expression is simplified:

[along the perpendicular to the middle of the dipole, for r >> l].

It can be seen that the electric field strength of the dipole decreases with distance faster than for a point charge (like 1/r 3 instead of 1/r 2). This is to be expected: at large distances, two charges of opposite signs seem so close that they cancel each other out. The dependence of the form 1/r 3 is also valid for points that do not lie on the perpendicular to the middle of the dipole.