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

The electrical capacity of a solitary conductor depends only on its shape and dimensions, as well as on the surrounding dielectric medium (e). The unit of capacitance in the SI system is called Farada. Farad (F) is the capacity of such a solitary conductor, whose potential rises by 1 Volt when it charges a charge of 1 Pendant. 1 F = 1 C / 1 V.

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

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

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

In a series connection, the charge on the plates of all capacitors is the same Q1 = Q2 = Q3, and the battery voltage is equal to the sum of the voltages of the individual capacitors U = U1 + U2 + U3.

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

The capacitance of a battery of series-connected capacitors is always less than the capacitance of each of these capacitors individually. The energy of the electrostatic field. The energy of a charged flat capacitor Ek is equal to the work A, which was spent during its charging, or is performed when it is discharged. A = CU2 / 2 = Q2 / 2C = QU / 2 = EK. Since the voltage across the capacitor can be calculated from the relation: U = E * d, where E is the field strength between the capacitor plates, d is the distance between the plates of the capacitor, then the energy of the charged capacitor is: Ek = CU2 / 2 = ee0S / 2d * E2 * d2 = ee0S * d * E2 / 2 = ee0V * E2 / 2, where V is the volume of the space between the plates of the capacitor. The energy of a charged capacitor is concentrated in its electric field.

Like any system of charged bodies, capacitor  has energy. Calculating the energy of a charged flat capacitor with a homogeneous field inside it is not difficult. The energy of a charged capacitor. In order to charge the capacitor, you need to perform work on the separation of positive and negative charges. According to the law of conservation of energy, this work is equal to the energy of the condenser. The fact that a charged capacitor has energy can be seen if it is discharged through a circuit containing an incandescent lamp rated at several volts fig.14.37). When the condenser is discharged, the lamp flashes. The energy of the condenser is converted into heat and light energy.

Let us derive the formula for the energy of a flat capacitor. The field strength created by the charge of one of the plates is E / 2, where E  is the field strength in the capacitor. In a homogeneous field of one plate there is a charge q, distributed over the surface of another plate ( fig.14.38). According to the formula (14.14) for the potential energy of a charge in a homogeneous field energy  The capacitor is:

where q  - charge of the capacitor, and d  is the distance between the plates. As Ed = U, where U  - the potential difference between the capacitor plates, then its energy is:

This energy is equal to the work done by the electric field when the plates are brought together closely. Replacing in (14.25) the potential difference or charge by means of the expression (14.22) for the capacitance of the capacitor, we obtain:

W = qU / 2 = q ^ 2 / 2C = CU ^ 2/2

One can prove that these formulas are valid for any condenser, and not just for a plane capacitor. Energy of electric field.According to the theory of short-range interaction, the entire energy of the interaction of charged bodies is concentrated in the electric field of these bodies. Hence, the energy can be expressed through the basic characteristic of the field - tensions. Since the electric field strength is directly proportional to the potential difference ( U = Ed, then according to the formula W = qU / 2 = q ^ 2 / 2C = CU ^ 2/2

the condenser energy is directly proportional to the square of the electric field inside it: W ~ E ^ 2. Application of capacitors. The dependence of the capacitance of the capacitor on the distance between its plates is used when creating one of the types of computer keyboards. On the back of each key there is one plate of the capacitor, and on the board, located under the keys, is the other. Pressing the key changes the capacity of the capacitor. An electronic circuit connected to this capacitor converts the signal to the appropriate code sent to the computer. The energy of the capacitor is usually not very large - not more than hundreds of joules. In addition, it does not persist for long because of the inevitable leakage of charge. Therefore, charged capacitors can not replace, for example, batteries as sources electrical  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 more or less long time, and when discharged through a circuit with low resistance they give energy almost instantly. It is this property that is widely used in practice. A flash lamp used in photo, is powered by the electric current of the discharge of the capacitor, which is charged beforehand by a special battery. Excitation of quantum light sources - lasers is carried out with the help of a gas discharge tube, the flash of which occurs when the battery of capacitors of large electrical capacity is discharged. However, capacitors are mainly used in radio engineering. The energy of the capacitor is proportional to its electrical capacity and the square of the voltage between the plates. All this energy is concentrated in an electric field. The energy of the field is proportional to the square of the field strength.

There is a solitary explorer. He is informed of the charge Q. Let us calculate the electric potential at the point M.

If a charge is placed on the conductor Q · b, then

The potential at each point of the field increases in direct proportion to the charge of the conductor, i.e. φ ~ Q.

, (2)

where c is the electric capacity (capacity)

Or one can show: ΔQ = cΔφ

(3)

The physical meaning of capacity.

Note that all the previous ones are correct, if the shape and dimensions of the conductor do not change, as well as the external conditions (environment, location of surrounding objects).

SI:

IV.Condensators. Calculation of the capacitance of capacitors.

Capacitoris a system of two (or more) conductors having such a form and location relative to each other that the field created by such a system is localized in a limited region of space.

Examples of capacitors:

The conductors forming the condenser are called plates.

To charge the capacitor, it is necessary to attach its plates to the voltage source or to connect one plate to the Earth, and the other ("+") to the source terminal.

The capacitance of the capacitor Cis a quantity measured by the ratio of the charge Q on one plate to the potential difference between the plates:

(4)

Examples of capacitor capacitance calculation.

1. Flat capacitor.

(5)

2. Spherical capacitor.

We put: r 1 -r 2 = d; d<

Consequence:

if the gap is small, then С пл = С сф

if r 1 \u003e\u003e r 2, then С сф = 4πεε 0 r → C сф = С of the ball

3. Cylindrical capacitor.

(7)

If the voltage Uon the capacitor to make it too large, then a discharge through the dielectric layer - a breakdown occurs. Therefore, each capacitor is characterized not only by its capacitance C, but also by the maximum operating voltage U max = U pr.

With capacitors of different capacities, the desired capacitance can be obtained by connecting capacitors:

V. Energy of a charged capacitor. Energy of an electric field.

For many questions of theory and practice, it is necessary to determine the electrical energy of a charged conductor. (We determine through the work of the discharge of the conductor).

Let there be a conductor with a charge Q and an initial potential φ 0. Then the elementary work in the transition of the elementary charge dQc of the conductor to the earth is equal to:

dA = φ · dQ, where

 - instantaneous potential, but

dQ = -Cdφ ("-" - means a decrease in the potential).

dA = -Cφ · dφ

The work was found due to the decrease in the potential energy and is numerically equal to the energy of the charged conductor W:

The formula for the energy of a charged body essentially determines the energy of the electric field created by a charged body:

(8)

Volumetric energy densityelectrostatic field - a physical quantity that is numerically equal to the ratio of the potential field energy per unit volume.

(9)

When creating circuits during the amateur radio electronics, you have to operate a significant amount of terminology. And one of the most important components are capacitors. In themselves, they are not very interesting, more important for us - their functions. Here, for example, the electric energy of the capacitor. What is it? It is due to the fact that the electric field, which is located between the plates of the capacitor, itself has energy. So, its tension is proportional to the applied voltage. Let's consider in more detail and with a number of formulas.

The energy of a charged capacitor

The capacitor plates have an electrical capacity (E). There are two electric charge:  -s and + s. Then the voltage (H), which exists between the plates, is equal to:

• H = s / E

All the components of this equation were considered above, and if you are confused, re-read until you can understand. Without this, it will be impossible to continue acquaintance with the material of the article, so that it can be learned. This knowledge is necessary to understand how the energy of the capacitor field functions.

But the device is discharged with time. What to do about it? When the discharge process takes place, the voltage existing between its plates will decrease in direct proportion to the charge from the initial value to zero. In the formula expression, this equation will look like this:

• Hp = H / 2 = s / 2 * E

But we still have work A, which is accomplished by an electric field during the discharge of the capacitor. In the formula view, everything looks like this:

• A = s * Hp = (s * H) / 2 = (E * H 2) / 2

But with this the question arises: what will be equal to potential energy  capacitor with a given electric capacity E, which is charged to the value of H? The answer to this question can give us such an equation:

• AE = A = (E * H 2) / 2 = s 2 / (2 * E) = (s * H) / 2

Here you should understand that the energy of the capacitor depends on the electric field, which exists between its plates, and it is its owner. And from this we can conclude that it is also proportional to the square of the tension. To remember what the energy of a charged capacitor is, you can learn one more school rule. Or even more precise will say - refresh your memory to them. The energy of the capacitor is equal to the work done by the electric field during the approach of the plates of the device closely. It is also equal to labor, which is done to separate the negative and positive charges, which is necessary for the subsequent charging of the device. This is studied as an example in the course of school physics.

Electric capacity

In the previous section of the article mentioned such a word. Given its importance, when analyzing the situation with a capacitor, you can figure out what is meant by this word. So, the electrical capacity:

1. Used as a characteristic of the ability to accumulate an electric charge by a capacitor.
2. It is dependent on a number of parameters:
   1. From the geometric dimensions of the capacitor.
   2. From its shape.
   3. From the location in the circuit.
   4. From the properties of the electrical environment, in which the condenser itself is located.
3. It does not depend on the values ​​of charge and voltage.

The electric capacity is measured in Farad (in practice, the micro-attachment is added, since the volume of the capacitor is usually small).

Field energy and formula

It roughly equals the square of the electric field inside the capacitor.

The energy density is measured by the formula:

What can we say about this further? This effect is added to the others and can constitute the electric field of the entire device, of which the capacitor is a part.

Conclusion

So, within the framework of the article, the energy of the capacitor, as well as the field created by it, was considered. It must also be taken into account that other parts of electrical circuits also have a certain energy and can positively affect the degree of charge of this device. If the capacitor is beyond the edges of the circuits and is not used by them, but is near, it will gradually be charged. Truthfulness of this fact is very easy to check at home if there is a necessary measuring technique. To do this, you must place the capacitor itself near a TV, radio or computer device and record the charge value that the measuring equipment will display. Due to this property, the energy of the capacitor can change even if there is no direct visible connection to the power supply.

The electrical capacitance of the conductor.

Electrical capacitance  - characteristics of the conductor, a measure of its ability to accumulate electric charge  . In the theory of electrical circuits, capacity refers to the mutual capacitance between two conductors; parameter of the capacitive element of the electrical circuit, represented as a two-terminal network. Such a capacitance is defined as the ratio of the electric charge to potential difference  between these conductors.

Coupling of capacitors.

When the capacitor is charged, the external source spends energy on the separation of charges into positive and negative charges. Which will be on the capacitor plates. Hence, proceeding from the law of conservation of energy, it does not disappear anywhere, but remains in the condenser. The energy in the condenser is stored in the form of a force of interaction of the positive and negative charges located on its plates. That is, in the form of an electric field. Which is concentrated between the plates. This interaction tends to draw one plate to the other, since, as is known, unlike charges are attracted.

As is known from mechanics   F = mg, likewise in an electrician F = qE, the role of the mass is played by the charge, and the role of the force of attraction is the intensity of the field.

The work on charge transfer in an electric field looks like this : A = qEd1-qEd2 = qEd

On the other hand, the work is also equal to the difference of potential energies A = W1-W2 = W.

Thus, using these two expressions, we can conclude that the potential energy stored in the capacitor is equal to:

Formula 1 - The energy of a charged capacitor

It is not difficult to see that the formula is very similar to the potential energy from mechanics W = mgh.

If we draw an analogy with mechanics: Imagine a stone on the roof of a building. Here, the mass of the earth interacts with the mass of the stone by gravity, and the height of the building   hcounteracts the force of gravity. If the building to remove the stone falls, therefore, the potential energy will go to the kinetic.

In electrostatics, however, there are two dissimilar charges of the aspirants attracted to each other, they are counteracted by a dielectric thickness dlocated between the plates. If the plates are closed between each other, then the potential energy of the charge goes to the kinetic energy, that is, to heat.

In electrical engineering, the formula for energy in this form is not applied. It is convenient to express it through the capacitance of the capacitor and the voltage to which it is charged.

Since the charge of the capacitor is determined by the charge of one of its plates, the intensity of the field created by it will be E / 2. Since the total field is composed of the fields created by both plates, charging is the same, but with the opposite sign.

Topics of the USE codifier: electric capacitance, capacitor, energy of the electric field of the capacitor.

The previous two articles were devoted to a separate examination of how conductors behave in the electric field and how dielectrics are used. Now we need to combine this knowledge. The fact is that the practical use of conductors and dielectrics in special devices is of great practical importance - capacitors.

But first we introduce the notion electrical capacity.

Capacitance of a solitary conductor

Suppose that a charged conductor is located so far from all other bodies that the interaction of the charges of the conductor with the surrounding bodies can be ignored. In this case, the conductor is called secluded.

The potential of all points of our conductor, as we know, has one and the same meaning, which is called the potential of the conductor. It turns out that the potential of a solitary conductor is directly proportional to its charge. The coefficient of proportionality is usually denoted, so that

The value is called electric capacity  conductor and is equal to the ratio of the conductor's charge to its potential:

(1)

For example, the potential of a solitary ball in a vacuum is:

where is the charge of the ball, is its radius. Hence the capacity of the ball:

(2)

If the sphere is surrounded by a dielectric medium with a dielectric constant, its potential decreases by a factor of:

Accordingly, the capacity of the ball increases by a factor of:

(3)

The increase in the capacity in the presence of a dielectric is the most important fact. We will meet with him again when considering capacitors.

From formulas (2) and (3) we see that the capacity of a sphere depends only on its radius and the dielectric constant of the surrounding medium. The same will happen in the general case: the capacity of a solitary conductor is independent of its charge; it is determined only by the dimensions and shape of the conductor, and also by the dielectric constant of the medium surrounding the conductor. The capacitance of the conductor material is also independent.

What is the meaning of the concept of capacity? The capacitance shows how much charge to inform the conductor in order to increase its potential on the B. The larger the capacitance, the higher the charge, respectively, for the conductor.

The unit of measurement of capacity serves farad  (Φ). From the definition of the capacity (1), it is clear that Φ = л /..

Let's calculate for sake of interest the capacity of the globe (it is the conductor!). We consider the radius to be approximately equal to km.

MkF.

As you can see, F is a very large capacity.

The unit of capacitance is also useful in that it makes it possible to save a lot on the designation of the dimension of the dielectric constant. In fact, we express from formula (2):

Consequently, the dielectric constant can be measured in F / m:

F.

It's easier to remember, is not it?

Capacitance of a flat capacitor

The capacity of a solitary conductor is rarely used in practice. In ordinary situations, conductors are not secluded. A charged conductor interacts with surrounding bodies and charges them, and the field potential of these induced charges (by the principle of superposition!) Changes the potential of the conductor itself. In this case, it can no longer be asserted that the potential of the conductor will be directly proportional to its charge, and the concept of capacitance of the conductor in itself actually loses its meaning.

It is possible, however, to create a system of charged conductors, which, even when a significant charge accumulates on them, almost does not interact with surrounding bodies. Then we can again talk about capacity - but this time about the capacity of this system of conductors.

The simplest and most important example of such a system is flat capacitor. It consists of two parallel metal plates (called plates), separated by a dielectric layer. The distance between the plates is much smaller than their own sizes.

Let's start by looking at air  condenser, which has air between the plates

Let the charges of the plates be equal and. This is the case in real electrical circuits: the charges of the plates are equal in absolute value and opposite in sign. The value - the charge of the positive electrode - is called charge of the capacitor.

Let be the area of ​​each plate. Find the field created by the plates in the surrounding space.

Since the dimensions of the plates are large compared to the distance between them, the field of each facing far from its edges can be considered a homogeneous field of an infinite charged plane:

Here - the strength of the positive electrode field, - the field strength of the negative electrode, - the surface density of charges on the plate:

In Fig. 1 (left) shows the field strength vectors of each electrode in three areas: to the left of the capacitor, inside the capacitor and to the right of the capacitor.

Fig. 1. The electric field of a flat capacitor

According to the principle of superposition, for the resulting field we have:

It is not difficult to see that the field vanishes to the left and right of the condenser (the field of the plates cancel each other):

Inside the capacitor, the field is doubled:

(4)

The resulting field of the plates of the flat capacitor is shown in Fig. 1 on the right. So:

Inside the flat capacitor a homogeneous electric field is created, the strength of which is found by the formula (4). From the outside of the capacitor, the field is zero, so that the capacitor does not interact with surrounding bodies.

Let's not forget, however, that this statement is derived from the assumption that the plates are infinite planes. In fact, their dimensions are finite, and near the edges of the plates there are so-called edge effects: the field differs from the homogeneous and penetrates into the outer space of the condenser. But in most situations (and even more so in USE physics problems), edge effects can be neglected and act as if the statement in italics is true without any reservations.

Let the distance between the capacitor plates equal. Since the field inside the capacitor is homogeneous, the potential difference between the plates is equal to the product on (recall the connection of voltage and voltage in a homogeneous field!):

(5)

The potential difference between the capacitor plates, as we see, is directly proportional to the capacitor charge. This statement is analogous to the statement "the potential of a solitary conductor is directly proportional to the charge of the conductor," from which the whole conversation about capacity began. Continuing this analogy, we define capacitor capacitance  as the ratio of the charge of the capacitor to the potential difference between its plates:

(6)

The capacity of the capacitor shows what charge it needs to communicate, so that the potential difference between its plates increases by B. Formula (6) is thus a modification of formula (1) for the case of a system of two conductors-a capacitor.

From formulas (6) and (5) we easily find capacity of flat air condenser:

(7)

It depends only on the geometric characteristics of the capacitor: the area of ​​the plates and the distance between them.
  Suppose now that the space between the plates is filled with a dielectric with dielectric permittivity. How will capacitance change?

The field strength inside the capacitor will decrease by a factor of one, so that instead of (4) we now have:

(8)

Accordingly, the voltage across the capacitor:

(9)

From here capacitance of a flat capacitor with a dielectric:

(10)

It depends on the geometric characteristics of the capacitor (the area of ​​the plates and the distance between them) and the dielectric constant of the dielectric that fills the capacitor.

An important consequence of formula (10): capacitor filling with a dielectric increases its capacitance.

The energy of a charged capacitor

A charged capacitor has energy. This can be seen from experience. If you charge the capacitor and close it to a light bulb, then (assuming that the capacitance of the capacitor is large enough), the light will briefly light up.

Consequently, the charged capacitor stores energy, which is released when it is discharged. It is easy to understand that this energy is the potential energy of interaction between the capacitor plates - after all, the plates, being charged in different ways, are attracted to each other.

We now calculate this energy, and then we see that there is a deeper understanding of the origin of the energy of the charged capacitor.

Let's start with a flat air condenser. Let's answer this question: what is the force of attraction of its plates to each other? The values ​​used are the same: the charge of the capacitor, the area of ​​the plates.

Take on the second plate so small a platform that the charge of this site can be considered a point. This charge is attracted to the first plate with a force

where is the strength of the first lining field:

Consequently,

This force is directed parallel to the lines of the field (that is, perpendicular to the plates).

The resultant force of attraction of the second plate to the first is composed of all these forces, with which all small charges of the second electrode are attracted to the first plate. In this summation, the constant factor will be taken outside the bracket, and all will be summed in the bracket and given. As a result, we get:

(11)

Suppose now that the distance between the plates has changed from the initial value to the final value. The force of attraction of the plates performs the following work:

The sign is correct: if the plates approach each other, the force performs positive work, since the plates are attracted to each other. Conversely, if you remove the plates\u003e, then the work of the attractive force turns out to be negative, as it should be.

Taking into account formulas (11) and (7), we have:

This can be rewritten as follows:

(12)

The work of the potential force of attraction of the plates turned out to be equal to the change with a minus sign of magnitude. This just means that - the potential energy of interaction between the plates, or charged capacitor energy.

Using the relation, from formula (12) it is possible to obtain two more formulas for the energy of the capacitor (check it yourself!):

(13)

(14)

Especially useful are the formulas (12) and (14).

Suppose now that the capacitor is filled with a dielectric with a dielectric constant. The force of the attraction of the plates decreases by a factor of one, and instead of (11) we obtain:

When calculating the work of the force, as it is not difficult to see, the value will enter the capacity, and formulas (12) - (14) will remain unchanged. The capacity of the capacitor in them will now be expressed by formula (10).

Thus, formulas (12) - (14) are universal: they are valid for both an air condenser and a capacitor with a dielectric.

Electric field energy

We promised that after calculating the energy of the capacitor we will give a deeper interpretation of the origin of this energy. Well, let's get started.

Let us consider an air condenser and transform formula (14) for its energy:

But - the volume of the condenser. We get:

(15)

Look carefully at this formula. It no longer contains anything that is specific to the capacitor! We see electric field energy  , concentrated in a certain volume.

The energy of the capacitor is none other than the energy of the electric field inside it.

So, the electric field itself has energy. There is nothing surprising for us here. Radio waves, sunlight are examples of the propagation of energy transported in space by electromagnetic waves.

The quantity-the energy per unit volume of the field-is called bulk density of energy. From the formula (15) we obtain:

(16)

In this formula, there are no geometric quantities left at all. It gives the most pure connection between the energy of the electric field and its intensity.

If the capacitor is filled with a dielectric, then its capacitance increases by a factor of one, and instead of formulas (15) and (16) we have:

(17)

(18)

As we see, the energy of the electric field also depends on the dielectric constant of the medium in which the field is located.
  It is remarkable that the formulas obtained for energy and energy density go far beyond electrostatics: they are valid not only for the electrostatic field, but also for electric fields that vary in time.