Volumetric energy density of the electric field. Volumetric energy density Volumetric energy density of a uniform electric field

In the case of real values, the volumetric energy density of the electromagnetic field is determined by the expression:

If we consider vectors and as vectors with complex components, then to obtain a real expression for the volumetric energy density of the electromagnetic field, it is necessary to use the technique described above:

Expression (8) determines the “instantaneous” value of the volumetric density of electromagnetic energy at the considered point in space, i.e. value at some point in time t. Dependence (8) is practically the sum of squares of real values ​​and therefore is a positive definite relationship. Its numerical values ​​can vary from zero to a certain maximum value. It is of interest to calculate the time-averaged volumetric energy density of the electromagnetic field of a plane wave. The time-average physical quantity is determined by the rule:

. (9)

For processes that are harmonic in time, the value is chosen equal to the oscillation period, and the reference point is chosen equal to zero.

It is easy to see that the following relations hold:

;

; (10)

.

Similar results are valid for magnetic field strength vectors.

Taking into account the results obtained, the time-average value of the volumetric energy density of the electromagnetic field at the considered point in space can be described by the dependence

Expression (11) is local, real and positive definite. Using it, you can calculate the energy of the electromagnetic field in a certain region of space:

, (12)

where the energy of the electric field and the energy of the magnetic field are defined by the relations

, . (13)

Integration in relations (13) is carried out over the volume of the considered region of space. These expressions will be used below in the analysis of balance energy relationships.

Umov-Poynting vector.

The energy flux density of the electromagnetic field, as is known, is determined by the expression

If it is necessary to use the results of the complex amplitude method, the real (real) expression for the vector is written in the form:

Evaluating the vector products in relation (15), we obtain:

;

.

.

As a result of time averaging of dependence (15) for the instantaneous value of the energy flux density vector, we arrive at the relation:

. (16)

In this way, a time-constant vector quantity with real components is obtained. It is interesting that - formally - the resulting expression is the real part of the complex expression

This gives rise to the possibility of introducing into consideration the “complex Umov-Poynting vector”:

. (18)

The feasibility of this technique is justified by the following ratio:

The physical content of relation (19) is that the time average of the electromagnetic field energy flux density vector in the harmonic approximation (a real constant vector quantity!) can be calculated as the real part of the complex Umov-Poynting vector.

Volumetric power density.

For real values, the volumetric power density is calculated by the expression

Expression (20) - the product of two harmonic quantities - is nonlinear, therefore, to obtain the real value in the complex amplitude method, it is necessary to proceed from the relationship:

Dependence (21) determines the real (real) value of the volumetric power density at an arbitrary point in time. Since the quantity under consideration oscillates in time, we can introduce a time-averaged value for the volumetric power density in the same way as was done above when considering the volumetric energy density:

Analysis of expression (22) shows that it is possible to introduce complex power density

since the relation is easy to check

. (24)

Now we can begin to consider the balance energy relationships in an inhomogeneous plane electromagnetic harmonic wave.

Complex analogue of Poynting's theorem.

Maxwell's equations - the equation of electromagnetic induction and the equation of total current in differential form - we write using the harmonic approximation:

Note that equations (25)-(26) are valid if the shape of the dependence of harmonic quantities on time is determined by relations (6).

If , then , since the first equation implies and . In other words, if a linear equation for a complex quantity is valid, then the complex conjugate equation is also valid. Let's use this mathematical statement and write equation (26) in complex conjugate form:

Let us multiply equation (25) scalarly by the vector, and equation (27) by the vector:

Let us subtract equation (29) from equation (28):

The left side of equation (30) can be transformed:

In principle, the well-known vector identity is used here; it can be verified by direct calculation in the Cartesian coordinate system, or you can use the symbolic method and the definition of the differential vector operator “nabla” (or Hamilton operator). Let's demonstrate this method. Consider the divergence of the vector product of two vector fields:

.

In order to be able to use the notation as a simple vector quantity, we rewrite the previous relationship taking into account the differential nature of the nabla operator:

where conditionally constant quantities are marked with the index “c”, they can be “carried out” beyond the symbol of the differential operator. Now the resulting expression can be considered simply as the sum of two mixed products of three vectors. It is known that the mixed product of three vectors can be written in several equivalent forms. We need to choose a form such that the “vector” does not remain in the rightmost position: as a differential operator, it must act on something.

According to basic physics, it is known that there is a magnetic field around a conductor or coil carrying current. This field fully depends on the conductor, the medium of propagation of the field and the current strength. Similar to the electric field, the magnetic field is a kind of carrier of energy. Since the main criterion affecting the field energy is the strength of the flowing current, the work done by the current to create a magnetic field will coincide with the energy of the magnetic field.

Magnetic field energy

It is easier to understand the nature of such a phenomenon as magnetic field energy by considering the processes taking place in the circuit.

Schematic elements:

1. L – inductor;
2. L – light bulb;
3. ε – direct current source;
4. K – key for closing and opening the circuit.

When the key is closed, according to picture (a), current flows from the positive terminal of the current source along parallel branches through the inductor and the light bulb. The inductor carries a current I0, and the light bulb carries a current I1. At the first moment of time, the light bulb will burn brighter due to the high resistance of the inductor. As the inductor resistance decreases and the current I0 increases, the light bulb will burn dimmer. This is explained by the fact that at the first moment of time the current entering the coil is proportional to the high frequency current, based on the formula of the inductive reactance of the coil:

XL=2πfL, where:

• XL – inductive reactance of the coil;
• f – current frequency;
• L – coil inductance.

The inductive reactance of the coil increases many times. The inductor at this point in time behaves like an open circuit. Over time, the inductive reactance decreases to zero. Since the active resistance of the inductor coil is negligible, and the resistance of the nichrome filament of the light bulb is high, almost the entire circuit current flows through the coil.

After opening the circuit with key K, according to picture (b), the light bulb does not go out, but, on the contrary, lights up with a brighter light and gradually goes out. Energy is required to burn a light bulb. This energy is taken from the magnetic field of the inductor and is called magnetic field energy. Thanks to this, the inductor acts as a source of energy (self-induction), according to picture (c).

It is possible to determine the activity of the magnetic field by examining the electrical circuit.

To calculate the energy of the magnetic field, there is a need to create a circuit in which the energy of the power source would be spent directly on the formation of the magnetic field. Accordingly, in the circuit above, the values ​​of the internal resistance of the power supply and inductor should be neglected.

Note! From Kirchhoff's second law it follows that the sum of the voltages connected to the circuit is equal to the sum of the voltage drops on each of the circuit elements.

The total voltage of the circuit is:

ε+εі=Ir+IR, where:

• ε – electromotive force (voltage) of the power source;
• εi – electromotive force (voltage) of induction;
• I – circuit current;
• r – internal resistance of the power source;
• R is the internal resistance of the inductor.

Since the considered circuit is ideal and the internal resistances are zero, the formula is transformed into this:

The electromotive force of self-induction depends on the inductance of the coil and the rate of change of current in the circuit, namely:

Substituting the value into the general formula, it turns out:

• ε-LΔI/Δt=0,
• ε= LΔI/Δt,
• ΔI= ε Δt /L.

Based on this pattern, over time the current strength is equal to:

The charge passed through the inductor is:

Combining both formulas, we get:

The work done by the current source to transfer charge through the inductor is equal to:

A= εq=εLI2/2ε=LI2/2.

Since the circuit under consideration is ideal, namely, there is no resistance, the work expended on the current source went to the formation of the magnetic field and corresponds to the energy of the magnetic field:

In order to eliminate the dependence of the magnetic field activity on the characteristics of the coil, it is necessary to transform the expression through the field characteristic, namely through the magnetic induction vector:

1. B=µ0µIn, where:

• B – vector of magnetic induction of the solenoid;
• µ0 – magnetic constant (µ0=4π×10-7 H/m)
• µ – magnetic permeability of the substance;
• I – current strength in the solenoid circuit;
• n is the winding density, (n=N/l, where N is the number of turns, l is the length of the solenoid).

1. L=µ0µn2V, where:

V is the volume of the coil (or the volume of the magnetic field concentrated in the coil) (V=Sl, S is the cross-sectional area of ​​the solenoid, l is the length of the solenoid).

If we use formulas (1 and 2), the expression defining the energy of the magnetic field looks like:

Wmag=B2V/2µ0µ.

The considered formula is valid provided that the background is of the same type. If the field is inhomogeneous, then it is necessary to consider a parameter characterizing the concentration of activity in this zone. This quantity is referred to as the volumetric magnetic field energy density.

Volumetric magnetic energy density

It is determined by the expression:

ωmag=Wmag/V, where:

• ωmag – volumetric magnetic field energy density;
• V is the volume of a certain zone where a magnetic field is created.

The unit of measurement of volumetric magnetic field energy density is the ratio – J/m3.

Substituting the value of the field energy into the desired expressionWmagician,we obtain the final formulation defining the bulk density:

ωmag= B2/2µ0µ.

The information presented reveals in detail the procedure for finding such a field parameter as the magnetic field energy. Since the indicated value is applicable for a uniform field, to carry out calculations in a non-uniform magnetic field, a value is used that determines the concentration or density of the field energy.

Video

If a conductor is placed in an external electrostatic field, then it will act on its charges, which will begin to move. This process proceeds very quickly, after its completion an equilibrium distribution of charges is established, in which the electrostatic field inside the conductor turns out to be zero. On the other hand, the absence of a field inside the conductor indicates the same potential value at any point of the conductor, and also that the field strength vector on the outer surface of the conductor is perpendicular to it. If this were not so, a component of the voltage vector would appear, directed tangentially to the surface of the conductor, which would cause the movement of charges, and the equilibrium distribution of charges would be disrupted.

If we charge a conductor located in an electrostatic field, then its charges will be located only on the outer surface, since, in accordance with Gauss’s theorem, due to the equality of the field strength inside the conductor to zero, the integral of the electric displacement vector will also be equal to zero D along a closed surface coinciding with the outer surface of the conductor, which, as was established earlier, must be equal to the charge inside the said surface, i.e. zero. This raises the question of whether we can impart any, no matter how large, charge to such a conductor. To answer this question, we will find the connection between the surface charge density and the strength of the external electrostatic field.

Let us choose an infinitesimal cylinder crossing the conductor-air boundary so that its axis is oriented along the vector E . Let us apply Gauss's theorem to this cylinder. It is clear that the flux of the electric displacement vector along the side surface of the cylinder will be zero due to the fact that the field strength inside the conductor is zero. Therefore the total flow of the vector D through the closed surface of the cylinder will be equal only to the flux through its base. This flow, equal to the product D∆S, Where ∆S– base area, equal to the total charge σ∆S inside the surface. In other words, D∆S = σ∆S, from which it follows that

D = σ, (3.1.43)

then the electrostatic field strength at the surface of the conductor

E = σ /(ε 0 ε) , (3.1.44)

Where ε – dielectric constant of the medium (air) that surrounds the conductor.

Since there is no field inside a charged conductor, creating a cavity inside it will not change anything, that is, it will not affect the configuration of the arrangement of charges on its surface. If a conductor with such a cavity is now grounded, then the potential at all points of the cavity will be zero. Based on this electrostatic protection measuring instruments from the influence of external electrostatic fields.

Now consider a conductor that is distant from other conductors, other charges and bodies. As we established earlier, the potential of a conductor is proportional to its charge. It was experimentally established that conductors made of different materials, being charged to the same charge, have different potentials φ . Conversely, conductors made of different materials that have the same potential have different charges. Therefore we can write that Q = Cφ, Where

C = Q/φ (3.1.45)

called electrical capacity(or simply capacity) solitary conductor. The unit of measurement for electrical capacitance is the farad (F), 1 F is the capacitance of such an isolated conductor, the potential of which changes by 1 V when a charge equal to 1 C is imparted to it.

Since, as was established earlier, the potential of a ball of radius R in a dielectric medium with dielectric constant ε

φ =(1/4πε 0)Q/εR, (3.1.46)

then taking into account 3.1.45 for the capacity of the ball we obtain the expression

C= 4πε 0 εR. (3.1.47)

From 3.1.47 it follows that a sphere in vacuum would have a capacity of 1 F and would have a radius of the order of 9*10 9 km, which is 1400 times the radius of the Earth. This suggests that 1 F is a very large electrical capacity. The Earth's capacitance, for example, is only about 0.7 mF. For this reason, in practice, millifarads (mF), microfarads (μF), nanofarads (nF) and even picofarads (pF) are used. Next, because ε is a dimensionless quantity, then from 3.1.47 we obtain that the dimension of the electrical constant ε 0 – F/m.

Expression 3.1.47 suggests that a conductor can have high capacity only if it is very large. In practical activities, devices are required that, with small sizes, would be capable of accumulating large charges at relatively low potentials, i.e., would have large capacities. Such devices are called capacitors.

We have already said that if a conductor or dielectric is brought closer to a charged conductor, charges will be induced on them so that charges of the opposite sign will appear on the side of the introduced body closest to the charged conductor. Such charges will weaken the field created by the charged conductor, and this will lower its potential. Then, in accordance with 3.1.45, we can talk about an increase in the capacity of the charged conductor. It is on this basis that capacitors are created.

Usually capacitor comprises two metal plates, separated dielectric. Its design should be such that the field is concentrated only between the plates. This requirement is satisfied two flat plates, two coaxial(having the same axis) cylinder different diameters and two concentric spheres. Therefore, capacitors built on such plates are called flat, cylindrical And spherical. In everyday practice, the first two types of capacitors are most often used.

Under capacitor capacity understand physical quantity WITH , which is equal to the charge ratio Q accumulated in the capacitor to the potential difference ( φ 1 – φ 2), i.e.

C = Q/(φ 1 – φ 2). (3.1.48)

Let us find the capacitance of a flat capacitor, which consists of two plates with an area S, separated from each other by a distance d and having charges +Q And –Q. If d is small compared to the linear dimensions of the plates, then edge effects can be neglected and the field between the plates can be considered uniform. Because the Q = σS, and, as was shown earlier, the potential difference between two oppositely charged plates with a dielectric between them φ 1 – φ 2 = (σ/ε 0 ε)d, then after substituting this expression into 3.1.48 we get

C= ε 0 εS/d. (3.1.49)

For a cylindrical capacitor with a length l and cylinder radii r 1 And r 2

C = 2πε 0 εl/ln(r 2 /r 1). (3.1.50)

From expressions 3.1.49 and 3.1.50 it is clearly seen how the capacitance of the capacitor can be increased. First of all, to fill the space between the plates, materials with the highest possible dielectric constant should be used. Another obvious way to increase the capacitance of a capacitor is to reduce the distance between the plates, but this method has an important limitation – dielectric breakdown, i.e., an electrical discharge through a dielectric layer. The potential difference at which electrical breakdown of a capacitor is observed is called breakdown voltage. This value is different for each type of dielectric. As for increasing the area of ​​flat plates and the length of cylindrical capacitors to increase their capacity, there are always purely practical limitations on the size of capacitors, most often these are the dimensions of the entire device that includes the capacitor or capacitors.

In order to be able to increase or decrease the capacitance, parallel or series connection of capacitors is widely used in practice. When capacitors are connected in parallel, the potential difference across the capacitor plates is the same and equal to φ 1 – φ 2, and the charges on them will be equal Q 1 = C 1 (φ 1 – φ 2), Q 2 = C 2 (φ 1 – φ 2), … Q n = C n (φ 1 – φ 2), so the battery is fully charged from the capacitors Q will be equal to the sum of the listed charges ∑Qi, which in turn is equal to the product of the potential difference (φ 1 – φ 2) to full capacity С = ∑Ci. Then for the total capacity of the capacitor bank we get

C = Q/(φ 1 – φ 2). (3.1.51)

In other words, when capacitors are connected in parallel, the total capacitance of the capacitor bank is equal to the sum of the capacitances of the individual capacitors.

When capacitors are connected in series, the charges on the plates are equal in magnitude, and the total potential difference ∆ φ batteries is equal to the sum of potential differences ∆ φ 1 at the terminals of individual capacitors. Since for each capacitor ∆ φ 1 = Q/C i, then ∆ φ = Q/C =Q ∑(1/C i), where we get

1/C = ∑(1/C i). (3.1.52)

Expression 3.1.52 means that when capacitors are connected in series into a battery, the reciprocal values ​​of the capacitances of individual capacitors are summed up, and the total capacitance turns out to be less than the smallest capacitance.

We have already said that the electrostatic field is potential. This means that any charge in such a field has potential energy. Let there be a conductor in a field for which the charge is known Q, capacity C and potential φ , and let us need to increase its charge by dQ. To do this you need to do work dA = φdQ = Сφdφ by transferring this charge from infinity to the conductor. If we need to charge the body from zero potential to φ , then you will have to do work that is equal to the integral of Сφdφ within the specified limits. It is clear that integration will give the following equation

A = Cφ 2 /2. (3.1.53)

This work goes towards increasing the energy of the conductor. Therefore, for the energy of a conductor in an electrostatic field we can write

W = Сφ 2 /2 = Q φ/2 = Q 2 /(2C). (3.1.54)

A capacitor, like a conductor, also has energy, which can be calculated using a formula similar to 3.1.55

W= С(∆φ) 2 /2 = Q∆φ/2 = Q 2 /(2C), (3.1.55)

Where ∆φ – potential difference between capacitor plates, Q is its charge, and WITH– capacity.

Let's substitute into 3.1.55 the expression for capacity 3.1.49 ( C= ε 0 εS/d) and take into account that the potential difference ∆φ = Ed, we get

W = (ε 0 εS/d)(Ed 2)/2 = ε 0 εE 2 V/2, (3.1.56)

Where V = Sd. Equation 3.1.56 shows that the energy of the capacitor is determined by the strength of the electrostatic field. From equation 3.1.56 we can obtain an expression for the volumetric density of the electrostatic field

w = W/V = ε 0 εE 2 /2. (3.1.57)

Control questions

1. Where are the electric charges located on a charged conductor?

2. What is the strength of the electrostatic field inside a charged conductor?

3. What does the electrostatic field strength at the surface of a charged conductor depend on?

4. How are devices protected from external electrostatic interference?

5. What is the electrical capacity of a conductor and what is its unit of measurement?

6. What devices are called capacitors? What types of capacitors are there?

7. What is meant by the capacitance of a capacitor?

8. What are the ways to increase the capacitance of a capacitor?

9. What is capacitor breakdown and breakdown voltage?

10. How is the capacitance of a capacitor bank calculated when capacitors are connected in parallel?

11. What is the capacitance of a capacitor bank when capacitors are connected in series?

12. How is the energy of a capacitor calculated?

Electric field energy.

The energy of charged conductors and capacitors is usually determined in terms of their charges and potentials. It is possible, however, to relate the energy of a charged system to the characteristics of its electric field. To do this, consider a flat capacitor, the parameters of which are indicated in Figure 52.1.

Let's use formula (51.5) and perform transformations taking into account expressions (41.2) and (35.3):

The value is the volume of space between the capacitor plates. Neglecting field distortions at the edges of the plates (edge ​​effect), we can assume that the field of the capacitor is concentrated between its plates. Then V- this is also the volume of the electric field. In accordance with this, we write formula (52.1) in the form

. (52.2)

Expression (52.2) determines the energy of a charged capacitor through the characteristics of the electric field: its strength E and volume V. Based on this, we can conclude that energy is localized in the electric field, that the field itself has energy, and not the electric charge. In this regard, it should be said that in electrostatics there is no answer to this question, since stationary fields created by electric charges are considered. Alternating fields can exist independently of electric charges and propagate in the form of electromagnetic waves. The transfer of energy by electromagnetic waves has been proven experimentally and is used in telecommunication systems. This gives grounds to assert that the electric field is a carrier of energy. Therefore, this equation determines the energy of the electric field. The connection between the field energy and its volume confirms the materiality of the electric field.

The value of energy per unit volume of the field is called volumetric energy density.

The field of a flat capacitor is uniform and the energy is distributed in it with the same density. Therefore we can write:

Unit of volumetric energy density - joule per meter cubed. Combining formulas (52.3) and (52.2), we obtain

.

Let's carry out the transformations using expression (47.1):

. (52.4)

Let's use the equation and replace the electrical bias in it D in accordance with formula (47.6):

. (52.5)

The first term in this expression coincides with the energy density of the electric field in vacuum (), the second term represents the energy expended on the polarization of the dielectric.

Formulas for energy density were obtained for a uniform field, but they are applicable for any field in an isotropic dielectric. This allows you to calculate the field energy contained in any volume:

, (52.6)

where for a non-uniform field the strength should be specified by the function .

Chapter 5. DC ELECTRIC CURRENT

1. Energy of a system of stationary point charges. Electrostatic interaction forces are conservative; therefore, the system of charges has potential energy. Let us find the potential energy of a system of two stationary point charges and located at a distance r from each other. Each of these charges in the field of the other has potential energy:

where and are, respectively, the potentials created by the charge at the point where the charge is located and by the charge at the point where the charge is located. According to formula (8.3.6),

By adding charges , , … to a system of two charges in succession, one can verify that in the case of n stationary charges, the interaction energy of the system of point charges is equal to

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

2. Energy of a charged solitary conductor. Let there be a solitary conductor whose charge, capacitance and potential are respectively equal to q, C, . Let's increase the charge of this conductor by dq. To do this, it is necessary to transfer the charge dq from infinity to an isolated conductor, spending work equal to

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

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

Formula (8.12.3.) 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 conductor potential equal to , from (8.12.1.) we find

where is the charge of the conductor.

3. Energy of a charged capacitor. Like any charged conductor, a capacitor has energy, which, in accordance with formula (8.12.3.), is equal to

where q is the charge of the capacitor, C is its capacity, and is the potential difference between the plates.

4. Electrostatic field energy. Let us transform formula (8.12.4.), which expresses the energy of a flat capacitor through charges and potentials, using the expression for the capacitance of a flat capacitor and the potential difference between its plates (). Then we get

where V=Sd is the volume of the capacitor. Formula (8.12.5.) shows that the energy of the capacitor is expressed through a quantity characterizing the electrostatic field - tension E.

Formulas (8.12.4.) and (8.12.5.) 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 time-constant fields of stationary charges, i.e. in it the fields and the charges that determine 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, capable transfer energy. This convincingly confirms the main point short-range theory of energy localization in a field So what carrier energy is field.

Bulk Density electrostatic field energy (energy per unit volume)

Expression (8.12.6.) is valid only for isotropic dielectric, for which the following relation holds: .