Let the potential of the capacitor plate on which the charge is located is equal to the potential of the plate on which the charge is located, Then each of the elementary charges into which the charge can be divided is at a point with potential and each of the charges into which the charge can be divided, at a point with potential .

According to formula (28.1), the energy of such a system of charges is equal to

Using relation (27.2), we can write three expressions for the energy of a charged capacitor:

Formulas (29.2) differ from formulas (28.3) only by replacing

Using the expression for potential energy, one can find the force with which the plates of a flat capacitor attract each other. Suppose that the distance between the plates can vary. We connect the beginning of the x axis with the left plate (Fig. 29.1). Then the x coordinate of the second plate will determine the gap d between the plates. According to formulas (27.3) and (29.2)

We differentiate this expression with respect to x, assuming that the charge on the plates is unchanged (the capacitor is disconnected from the voltage source). As a result, we obtain the projection onto the x axis of the force acting on the right plate:

The module of this expression gives the magnitude of the force with which the plates attract each other:

Now let’s try to calculate the attractive force between the plates of a flat capacitor as the product of the field strength created by one of the plates and the charge concentrated on the other. According to formula (14.3), the field strength created by one lining is equal to

The dielectric weakens the field in the gap by a factor of one, but this takes place only inside the dielectric (see formula (20.2) and the text associated with it). The charges on the plates are located outside the dielectric and therefore are under the action of the field of tension (29.4).

Multiplying the charge of the plate q by this tension, we obtain for the expression

Formulas (29.3) and (29.5) do not coincide. The value of force (29.3), obtained from the expression for energy, is consistent with experience. This is explained by the fact that, in addition to the “electric” force (29.5), mechanical forces are acting on the plates from the side of the insulator, trying to push them apart (see § 22; we note that we mean a liquid or gaseous dielectric). At the edge of the plates there is a scattered field that decreases in magnitude with distance from the edges (Fig. 29.2). Having a dipole moment, dielectric molecules experience the action of a force pulling them into the region of a stronger field (see formula (9.16)). As a result, the pressure between the plates increases and a force appears that weakens the action of the force (29.5) at a time.

If a partially charged capacitor with an air gap is partially immersed in a liquid dielectric, the dielectric is drawn into the space between the plates (Fig. 29.3). This phenomenon is explained as follows. - The dielectric constant of air is almost equal to unity. Therefore, before immersion of the plates in the dielectric, the capacitor capacity can be considered equal and the energy equal. When the gap is partially filled with a dielectric, the capacitor can be considered as two parallel-connected capacitors, one of which has a cladding area, equal to the relative part of the gap filled with liquid), and filled with a dielectric with the second with an air gap has a lining area equal to When capacitors are connected in parallel, the capacities add up:

Since the energy will be less than (the charge q is assumed to be unchanged - before immersion in the liquid, the capacitor was disconnected from the voltage source). Therefore, filling the gap with a dielectric is energetically beneficial. Therefore, the dielectric is drawn into the capacitor and its level in the gap rises. This in turn leads to an increase in the potential energy of the dielectric in the field of gravity. Ultimately, the dielectric level in the gap will be established at a certain height corresponding to the minimum of total energy (electric and gravitational). The phenomenon under consideration is similar to the capillary rise of a liquid in a narrow gap between the plates (see § 119 of the 1st volume).

The drawing of the dielectric into the gap between the plates can also be clarified from a microscopic point of view. There is an inhomogeneous field at the edges of the capacitor plates. Dielectric molecules have their own dipole moment or acquire it under the influence of a field; therefore, they are affected by forces tending to move them to the field of a strong field, i.e., inside the capacitor. Under the influence of these forces, the liquid is drawn into the gap until the electric forces acting on the liquid at the edge of the plates are balanced by the weight of the liquid column.

In a charged capacitor, the plates have opposite charges and interact with each other due to the electric field, which is concentrated in the space between the plates. The bodies between which there is an interaction are said to have potential energy. Therefore, we can talk about charged capacitor energy.

The plates of the charged capacitor interact with each other.

Availability energy  a charged condenser can be confirmed by experiments.

We take a capacitor of sufficiently large capacity, a current source, an on-heat lamp, and compose an electric circuit, the circuit of which is shown in Fig. 4.82. Translate the switch S  to position 1 and charge the capacitor to a certain potential difference from the source GB.  If after that, switch the switch to position 2, then you can observe a short flash of light due to the filament of the llama kidney. The observed phenomenon can be explained by the fact that the charged capacitor had the energydue to which the work was done on the glow of a spiral bulb.

In accordance with energy conservation law  the work performed when discharging the capacitor is equal to the work performed when charging it. The calculation of this work and, accordingly, the potential energy of the capacitor is complicated by the features of the capacitor charging process. Its plates are charged and discharged gradually. Charge dependence Q  capacitor versus time during charging is shown in the graph (Fig. 4.83). The charge not only increases gradually, but its rate of change does not remain constant. So, to carry out calculations based on the formula A \u003dqEd impossible, because the electric field does not remain constant. The potential difference also varies from zero to the maximum value. In fig. 4.84 it is shown that the potential difference varies in proportion to the charge of the capacitor. Such a dependence is characteristic of the elastic force, which depends on the elongation of the spring (Fig. 4.85).

Using this similarity, we can conclude that charged capacitor energy  will be equal

W \u003dQΔφ / 2.   Material from the site

This energy  equal to the work on charging the capacitor, which is numerically equal to the area of \u200b\u200bthe shaded triangle in the graph of Fig. 4.84.

Given that Q \u003d CΔφ we get

W \u003dC (Δφ) 2 / 2.

And if we take into account the connection of the potential difference with the charge Δφ = Q /C, then the potential energy of the capacitor can be calculated by the formula

W \u003d (Q / 2). (Q /C) \u003dQ 2/2C.

On this page, material on the topics:

• Spur charged capacitor energy

• Charged Capacitor Energy

• What physical quantities determine the energy of a capacitor

• Independent work on the electric capacity of a flat capacitor

• Yak viznachiti energy capacitor for another graph

Questions about this material:

The electrical capacity (capacity) C of a solitary insulated conductor is a physical quantity equal to the ratio of the change in the charge of the conductor q to the change in its potential f: C \u003d Dq / Df.

The electric capacity of a solitary conductor depends only on its shape and size, as well as on the surrounding dielectric medium (e). The unit of measurement of capacitance in the SI system is called Farada. Farada (F) is the capacity of such a solitary conductor, the potential of which increases by 1 Volt when a charge of 1 Coulomb is communicated to it. 1 F \u003d 1 C / 1 V.

A capacitor is a system of two oppositely charged conductors separated by a dielectric (for example, air). The property of capacitors to accumulate and store electric charges and the associated electric field is characterized by a value called the electric capacity of the capacitor. The electric capacity of the capacitor is equal to the ratio of the charge of one of the plates Q to the voltage between them U: C \u003d Q / U.

Depending on the shape of the plates, the capacitors are flat, spherical and cylindrical. The formulas for calculating the capacitances of these capacitors are given in the table.

Connect capacitors to batteries. In practice, capacitors are often connected to batteries - in series or in parallel.

With a parallel connection, the voltage on all plates is the same U1 \u003d U2 \u003d U3 \u003d U \u003d e, and the battery capacity is equal to the sum of the capacitances of the individual capacitors C \u003d C1 + C2 + C3.

When connected in series, the charge on the plates of all capacitors is the same Q1 \u003d Q2 \u003d Q3, and the battery voltage is equal to the sum of the voltages of the individual capacitors U \u003d U1 + U2 + U3.

The capacity of the entire system of series-connected capacitors is calculated from the ratio: 1 / C \u003d U / Q \u003d 1 / C1 + 1 / C2 + 1 / C3.

The battery capacity of series-connected capacitors is always less than the capacity of each of these capacitors individually. The energy of an electrostatic field. The energy of a charged flat capacitor Ek is equal to the work A, which was expended during its charging, or is accomplished during its discharge. A \u003d CU2 / 2 \u003d Q2 / 2C \u003d QU / 2 \u003d Ek. Since the voltage on the capacitor can be calculated from the relation: U \u003d E * d, where E is the field strength between the capacitor plates, d is the distance between the capacitor plates, the energy of the charged capacitor is equal to: Eк \u003d CU2 / 2 \u003d ee0S / 2d * E2 * d2 \u003d ee0S * d * E2 / 2 \u003d ee0V * E2 / 2, where V is the volume of space between the capacitor plates. The energy of a charged capacitor is concentrated in its electric field.

Like any charged body system, capacitor  possesses energy. It is not difficult to calculate the energy of a charged flat capacitor with a uniform field inside it. The energy of a charged capacitor.  In order to charge the capacitor, you need to do the work of separating positive and negative charges. According to the law of conservation of energy, this work is equal to the energy of the capacitor. The fact that a charged capacitor has energy can be verified by discharging it through a circuit containing an incandescent lamp, designed for a voltage of several volts ( fig. 14.37) When the capacitor is discharged, the lamp flashes. The energy of a capacitor is converted into heat and the energy of light.

We derive the formula for the energy of a flat capacitor. The field strength created by the charge of one of the plates is E / 2where E  - field strength in the capacitor. In a uniform field of one plate there is a charge qdistributed over the surface of another plate ( fig. 14.38) According to formula (14.14) for the potential charge energy in a uniform field energy  capacitor is equal to:

where q  is the charge of the capacitor, and d  - the distance between the plates. Because Ed \u003d uwhere U  - the potential difference between the capacitor plates, then its energy is equal to:

This energy is equal to the work that the electric field will perform when the plates come close together. Replacing in the formula (14.25) the potential difference or charge using the expression (14.22) for the electric capacity of the capacitor, we obtain:

W \u003d qU / 2 \u003d q ^ 2 / 2C \u003d CU ^ 2/2

It can be proved that these formulas are valid for any capacitor, and not just for a flat one. The energy of the electric field.According to the theory of close range, all the interaction energy of charged bodies is concentrated in the electric field of these bodies. So, energy can be expressed through the main characteristic of the field - tension. Since the electric field is directly proportional to the potential difference ( U \u003d ed, then according to the formula W \u003d qU / 2 \u003d q ^ 2 / 2C \u003d CU ^ 2/2

the capacitor energy is directly proportional to the square of the electric field inside it: W ~ E ^ 2. Capacitor Application. The dependence of the capacitance on the distance between its plates is used to create one of the types of computer keyboards. There is one capacitor plate on the back of each key, and another on the board located under the keys. Pressing a key changes the capacitance of the capacitor. An electronic circuit connected to this capacitor converts the signal into the corresponding code transmitted to the computer. The energy of the capacitor is usually not very large - no more than hundreds of joules. Moreover, it does not persist for a long time due to the inevitable charge leakage. Therefore, charged capacitors cannot replace, for example, batteries as sources electric  energy. But this does not mean at all that capacitors as energy storage devices have not received practical application. They have one important property: capacitors can accumulate energy for a more or less long time, and when discharged through a circuit with low resistance, they give off energy almost instantly. This property is widely used in practice. Flash lamp used in photo, is powered by an electric current discharge capacitor, previously charged with a special battery. The excitation of quantum light sources - lasers is carried out using a gas discharge tube, the flash of which occurs when the capacitor bank of large electric capacity is discharged. However, the main application of capacitors is in radio engineering. The energy of a capacitor is proportional to its electric intensity and the square of the voltage between the plates. All this energy is concentrated in an electric field. The field energy is proportional to the square of the field strength.