Electric capacity. Electrical capacity units. Capacitors. What is the electrical capacity of a capacitor? The electrical capacity of a flat capacitor will change if you change

The formula for electrical capacity is as follows.

This value is measured in farads. Typically, cell capacitance is very small and is measured in picofarads.

In problems, it is often asked how the electrical capacity of a capacitor will change if the charge or voltage is increased. This is a trick question. Let's draw another analogy.

Imagine that we are talking about an ordinary jar, not a capacitor. For example, you have a three-liter one. A similar question: what happens to the capacity of a can if you pour 4 liters of water into it? Of course, the water will just pour out, but the size of the can will not change in any way.

It's the same with capacitors. Charge and voltage have no effect on capacity. This parameter depends only on the actual physical dimensions.

The formula will be as follows

Only these parameters affect the real electrical capacity of the capacitor.

Any capacitor is marked with technical parameters.

It's easy to figure it out. A minimal knowledge of electricity is enough.

Connecting capacitors

Capacitors, like resistances, can be connected in series and in parallel. In addition, there are mixed compounds in the schemes.

As you can see, the electrical capacity of the capacitor is calculated differently in both cases. This also applies to voltage and charge. The formulas show that the electrical capacity of the capacitor, or rather, their combination in the circuit, will be the largest when connected in parallel. With sequential, the total capacity is significantly reduced.

When connected in series, the charge is distributed evenly. It will be the same everywhere - both total and on each capacitor. And when the connection is parallel, the total charge is added. This is important to remember when solving problems.

The voltage is considered the other way around. With a serial connection, we add, and with a parallel connection, it is equal everywhere.

Here you have to choose: if you need more voltage, then we sacrifice capacity. If the capacitance, then there will be no huge voltage.

Types of capacitors

There are a huge number of capacitors. They differ in both size and shape.

Of course, capacity is calculated differently for everyone.

Electric capacity of a flat capacitor

The electrical capacity of a flat capacitor is the easiest to determine. This formula is generally remembered by everyone, unlike others.

It all depends on the physical parameters and the environment between the plates.

It also matters which dielectric or material is placed inside. Since the part has the size of a sphere, its capacity depends on the radius.

In the case of a cylindrical shape, in addition to the medium inside, the radii and the length of the cylinder matter.

Think about how the electrical capacity of a flat capacitor will change if it is damaged? There are various malfunctions that can affect the performance of the capacitors.

For example, they dry out or swell. After that, they become unsuitable for the normal operation of the device where they are installed.

Consider examples of damage and failure of capacitors. Everyone can swell at once.

Sometimes only a few fail. This happens when capacitors of different parameters or quality.

An illustrative example of spoilage (bloating, tearing and escaping the contents).

If you see tapes like this, it's extreme damage. It couldn't be worse.

If you notice such swollen capacitors on a device (for example, a video card in a computer), this is a reason to think about replacing a part.

Such problems can only be eliminated by replacing it with a similar part. You must match all parameters one to one. Otherwise, the work may be incorrect or very short-term.

Capacitors must be changed carefully without damaging the board. You need to solder quickly, avoiding overheating. If you do not know how to do this, it is better to take the part for repair.

The main cause of destruction is overheating, which occurs in the event of aging or high resistance in the circuit.

It is recommended not to delay the repair. Since damaged capacitors change their capacitance, the device where they are located will operate abnormally. And over time, this can cause failure.

If you have swollen capacitors on your video card, then their timely replacement can correct the situation. Otherwise, the microcircuit or something else may burn out. In this case, repairs will be very expensive or even impossible.

Precautions

Above was an example with a can of water. It said that if you pour more water, the water will pour out. Now think about where the electrons in the capacitor can "pour out"? After all, it is completely sealed!

If you apply more current to the circuit than the capacitor is designed for, then as soon as it is charged, its excess will try to go somewhere. And there is no free space. The result will be an explosion. If the charge is slightly exceeded, the cotton will be small. But if you apply a huge amount of electrons to a capacitor, it will simply burst, and the dielectric will flow out.

Be careful!

Flat capacitor usually called a system of flat conducting plates - plates separated by a dielectric. The simplicity of the design of such a capacitor makes it relatively easy to calculate its electrical capacity and obtain values \u200b\u200bthat coincide with the experimental results.

We will fix two metal plates on insulating supports and connect them to the electrometer so that one of the plates will be connected to the electrometer rod, and the other to its metal body (Fig. 4.71). With this connection, the electrometer will measure the potential difference between the plates, which form a flat capacitor of two plates. When conducting research, it is necessary to remember that

at a constant value of the charge of the plates, a decrease in the potential difference indicates an increase in the capacitor's electrical capacity, and vice versa.

Let us inform the plates of opposite charges and note the deviation of the electrometer needle. Bringing the plates closer to each other (decreasing the distance between them), we note a decrease in the potential difference. Thus, with a decrease in the distance between the plates of the capacitor, its electrical capacity increases. With increasing distance, the readings of the electrometer arrow increase, which is evidence of a decrease in electrical capacity.

inversely proportional to the distance between its plates.

C ~ 1 / d,

where d - distance between plates.

This dependence can be depicted by a graph of inverse proportional dependence (Fig. 4.72).

We will shift the plates one relative to the other in parallel planes, without changing the distance between them.

In this case, the overlapping area of \u200b\u200bthe plates will decrease (Fig. 4.73). An increase in the potential difference, noted by an electrometer, will indicate a decrease in electrical capacity.

An increase in the overlap area of \u200b\u200bthe bed will increase the capacity.

Electric capacity of a flat capacitor is proportional to the area of \u200b\u200bthe plates that overlap.

C ~S,

where S - the area of \u200b\u200bthe plates.

This dependence can be represented by a graph of direct proportional dependence (Fig. 4.74).

Having returned the plates to their initial position, we introduce a flat dielectric into the space between them. The electrometer will note a decrease in the potential difference between the plates, which indicates an increase in the capacitance of the capacitor. If another dielectric is placed between the plates, then the change in electrical capacity will be different.

Electric capacity of a flat capacitor Depends on the dielectric constant of the dielectric.

C ~ ε ,

where ε - dielectric constant of a dielectric. Material from the site

This dependence is shown in the graph in Fig. 4.75.

The results of the experiments can be summarized in the form formulas for the capacity of a flat capacitor:

C \u003dεε 0 S /d,

where S - plate area; d - the distance between them; ε - dielectric constant of the dielectric; ε 0 - electrical constant.

Capacitors, which consist of two plates, are very rarely used in practice. Typically, capacitors have many plates connected in a specific pattern.

On this page material on topics:

Solving problems on the subject of electrical capacity of a flat capacitor
How does a dielectric affect electrical capacity?
Flat capacitor theory
The graph of the electrical capacity of a flat capacitor from the area of \u200b\u200bits plates
Conclusion on electrical capacity

Questions about this material:

What is the structure of a flat capacitor?
By changing what value in the experiment, you can make a conclusion about the change in electrical capacity?

Consider two charged conductors. Let us assume that all lines of force starting on one of them end on the other. For this, of course, they must have equal and opposite charges. Such a system of two conducting bodies is called a capacitor.

Examples of capacitors. Examples of capacitors are two concentric conducting spheres (spherical, or ball, capacitor), two parallel flat conducting plates, provided that the distance between them is small compared to the dimensions of the plates (flat capacitor), two coaxial conducting cylinders, provided that their length large compared to the gap between the cylinders (cylindrical condenser).

The two conductors that make up the capacitor are called the plates.

Figure: 41. Electric field in spherical, flat and cylindrical capacitors

In all such systems, when charges of equal magnitude and opposite in sign are imparted to the plates, the electric field is almost entirely contained in the space between the plates (Fig. 41). The appearance of some capacitors used in technology is shown in Fig. 42.

The main characteristic of a capacitor is electrical capacity or simply capacitance C, defined as the ratio of the charge of one of

plates to the potential difference, i.e. to the voltage, between them:

The distribution of charges on the plates will be the same regardless of whether a large or small charge is imparted to them. This means that the field strength, and therefore the potential difference between the plates, is proportional to the charge imparted to the capacitor. Therefore, the capacitance of a capacitor does not depend on its charge.

Figure: 42. Device, appearance and symbols on the electrical circuits of some capacitors

In a vacuum, the capacitance is determined solely by the geometric characteristics of the capacitor, that is, by the shape, size and mutual arrangement of the plates.

Capacity units. In SI, a farad is adopted as a unit of electrical capacity.Capacitance of 1 F is possessed by a capacitor, between the plates of which a voltage of 1 V is set when a charge of 1 C is communicated:

In the absolute electrostatic system of units of the CGSE, the electrical capacity has the dimension of length and is measured in centimeters:

In practice, one usually has to deal with capacitors whose capacitance is much less than 1 F. Therefore, fractions of this unit are used - microfarad (μF) and picofarad. The ratio between farad and centimeter is easy to establish given that

Capacitance and geometry of the capacitor. The dependence of the capacitance of a capacitor on its geometric characteristics can be easily illustrated by simple experiments. For this we will use an electrometer connected to two flat plates, the distance between which can be changed (Fig. 43). So that the charges of the plates are the same and the entire field is concentrated only between them, the second plate and the body of the electrometer should be grounded. The deflection of the electrometer needle is proportional to the voltage between the plates. If we move or move apart the capacitor plates, then with a constant charge, the voltage will accordingly decrease or increase: the capacitance is the greater, the smaller the distance between the plates. Similarly, you can make sure that the capacitance of the capacitor is the greater, the larger the area of \u200b\u200bits plates. To do this, you can simply move the plates with the same gap between them.

Figure: 43. The capacitance of the capacitor depends on the distance between the plates

Capacity of a flat capacitor. We obtain the formula for the capacity of a flat capacitor. The field between its plates is uniform except for a small area near the edges of the plates. Therefore, the voltage between the plates is equal to the product of the field strength E at the distance between them: To find the field strength E, you can use the formula (1) § 6, which connects E near the surface of the conductor with the surface charge density c: Let us express a through the charge of the capacitor and the area of \u200b\u200bthe plate, counting the distribution of the charge is uniform, which is consistent with the used assumption about the homogeneity of the field: Substituting the above relations into the general definition of capacitance (1), we find

In SI, where the capacitance of a flat capacitor has the form

In the CGSE system of units, k \u003d 1 and

The capacity of a spherical capacitor. In exactly the same way, we can derive a formula for the capacity of a spherical capacitor, considering the electric field in the gap between two charged concentric spheres of radii.The field strength there is the same as in the case of a solitary charged ball of radius.Therefore, the voltage between the plates of radii is

The expression for the capacity is obtained by substituting into the formula (1):

The capacity of a solitary conductor. Sometimes the concept of the capacitance of a solitary conductor is introduced, considering the limiting case of a capacitor, one of the plates of which is removed to infinity. In particular, the capacitance of a solitary conducting ball is obtained from (5) as a result of the passage to the limit, which corresponds to an unlimited increase in the radius of the outer plate with a constant radius of the inner

In the CGSE system of units, where the capacity of a solitary sphere is equal to its radius. If the conductor has a non-spherical shape, its capacitance is equal in order of magnitude to the characteristic linear size, although, of course, it also depends on its shape. Unlike a solitary conductor, the capacitance of a capacitor is much larger than its linear dimensions. For example, for a flat capacitor, the characteristic linear size is equal to where.As can be seen from formula (4), in this case

Dielectric capacitor. In the above examples of capacitors, the space between the plates was considered empty. Nevertheless, the obtained expressions for the capacity are also valid when this space is filled with air, as it was in the described simple experiments. If the space between the plates is filled with some kind of dielectric, the capacitance of the capacitor increases. This can be easily verified experimentally by pushing a dielectric plate into the gap between the plates of a charged capacitor connected to an electrometer (Fig. 43). With a constant charge of the capacitor, the voltage between the plates decreases, which indicates an increase in capacitance.

A decrease in the potential difference between the plates when a dielectric plate is introduced there indicates that the electric field strength in the gap becomes less. This decrease depends on what kind of dielectric is used in the experiment.

The dielectric constant. To characterize the electrical properties of a dielectric, a physical quantity is introduced, called the dielectric constant. Dielectric constant is a dimensionless quantity that shows how many times the electric field strength in a capacitor filled with a dielectric (or the voltage between its plates) is less than in the absence of a dielectric with the same capacitor charge. In other words, the dielectric constant shows how many times the capacitance of a capacitor increases when it is filled with a dielectric. For example, the capacitance of a flat capacitor filled with a dielectric with permittivity is

The definition of the permittivity given here corresponds to the phenomenological approach, in which only the macroscopic properties of matter in an electric field are considered. The microscopic approach, based on the consideration of the polarization of atoms or molecules that make up a substance, assumes the study of a specific model and allows not only to describe in detail the electric and magnetic fields inside the substance, but also to understand how macroscopic electric and magnetic phenomena occur in the substance. At this stage, we restrict ourselves only to the phenomenological approach.

Figure: 44. Parallel connection of capacitors

For solid dielectrics, the value ranges from 4 to 7, and for liquid - from 2 to 81. Ordinary pure water has such an abnormally high dielectric constant. In addition to an air capacitor of variable capacity (see Fig. 42) used to tune radio receivers, all other capacitors used in technology are filled with a dielectric.

Capacitor banks. When using capacitors, they are sometimes connected to form batteries. When connected in parallel (Fig. 44), the voltages across the capacitors are the same, and the total charge of the battery is equal to the sum of the charges of the capacitors for each of which, obviously, it is fair Considering the battery as one

capacitor, we have

On the other hand,

Comparing (8) and (9), we find that the capacity of the battery of parallel-connected capacitors is equal to the sum of their capacities:

Figure: 45. Series connection of capacitors

With a series connection of previously uncharged capacitors (Fig. 45), the charges on all capacitors are the same, and the total voltage is equal to the sum of the voltages on individual capacitors:

On the other hand, considering the battery as one capacitor, we have

Comparing (11) and (12), we see that when capacitors are connected in series, the values \u200b\u200binverse to the capacitances are added:

When connected in series, the battery capacity is less than the smallest of the capacitors connected.

When do two conductive bodies form a capacitor?

What is called a capacitor charge?

How to establish a connection between units of capacity SI and CGSE?

Explain qualitatively why the capacitance of a capacitor increases as the gap between the plates decreases.

Get a formula for the capacity of a flat capacitor, considering the electric field in it as a superposition of fields created by two oppositely charged planes.

Get a formula for the capacity of a flat capacitor, considering it as the limiting case of a spherical capacitor, in which they tend to infinity so that the difference remains constant.

Why can't we talk about the capacity of a solitary infinite flat plate or a separate infinitely long cylinder?

Briefly describe the difference between the phenomenological and microscopic approaches to the study of the properties of matter in an electric field.

What is the meaning of the dielectric constant of a substance?

Why, when calculating the capacity of the battery of series-connected capacitors, was it stipulated that they were not previously charged?

What is the point of connecting capacitors in series if it only leads to a decrease in capacitance?

Field inside and outside the capacitor. To emphasize the difference between what is called the charge of a capacitor and the total charge of the plates, consider the following example. Let the outer plate of the spherical capacitor be grounded, and the inner one is given a charge d. All this charge will be evenly distributed over the outer surface of the inner plate. Then a charge is induced on the inner surface of the outer sphere, therefore, the charge of the capacitor is equal. And what will happen on the outer surface of the outer sphere? It depends on what surrounds the capacitor. Let, for example, there is a point charge at a distance from the surface of the outer sphere (Fig. 46). This charge will in no way affect the electrical state of the internal space of the capacitor, i.e., the field between its plates. Indeed, the inner and outer spaces are separated by the thickness of the metal of the outer plate, in which the electric field is equal to zero.

Figure: 46. \u200b\u200bSpherical capacitor in an external electric field

Charge on the outer surface of the plate. But the nature of the field in external space and the charge induced on the outer surface of the outer sphere depend on the magnitude and position of the charge.This field will be exactly the same as in the case when the charge is located at a distance from the surface of a solid grounded metal ball, the radius of which is equal to the radius the outer sphere of the condenser (fig. 47). The induced charge will be the same.

To find the magnitude of the induced charge, we will argue as follows. An electric field at any point in space is created by a charge and a charge induced

on the surface of the ball, which is distributed there, of course, unevenly - just so that the resulting field strength inside the ball vanishes. According to the principle of superposition, the potential at any point can be sought in the form of the sum of the potentials of the fields created by a point charge and point charges, into which the induced charge distributed over the surface of the ball can be split. Since all elementary charges into which the charge induced on the surface of the ball is broken are at the same distance from the center of the ball, the potential of the field created by it in the center of the ball will be equal to

Figure: 47. The field of a point charge near a grounded conducting ball

Then the total potential in the center of the grounded ball is

The minus sign reflects the fact that the induced charge is always of the opposite sign.

So, we see that the charge on the outer surface of the outer sphere of the capacitor is determined by the environment in which the capacitor is located, and has nothing to do with the charge of the capacitor e. The total charge of the outer plate of the capacitor, of course, is equal to the sum of the charges of its outer and inner surfaces, however the charge of the capacitor is determined only by the charge of the inner surface of this plate, which is connected by the field lines of force with the charge of the inner plate.

In the considered example, the independence of the electric field in the space between the capacitor plates and, therefore, its capacitance from external bodies (both charged and uncharged) is due to electrostatic protection, i.e., the thickness of the metal of the outer plate. What the lack of such protection can lead to can be seen in the following example.

Flat capacitor with screen. Consider a flat capacitor in the form of two parallel metal plates, the electric field of which is almost entirely concentrated in the space between the plates. We enclose the capacitor in an uncharged flat metal box, as shown in Fig. 48. At first glance, it may seem that the picture of the field between the plates of the capacitor will not change, since the entire field is concentrated between the plates, and we neglect the edge effect. However, it is easy to see that this is not the case. Outside the capacitor, the field strength is zero, therefore, at all points to the left of the capacitor, the potential is the same and coincides with the potential of the left plate. In the same way, the potential of any point to the right of the capacitor coincides with the potential of the right plate (Fig. 49). Therefore, enclosing the capacitor in a metal box, we connect the points with different potential with a conductor.

As a result, a redistribution of charges will occur in the metal box until the potentials of all its points are equal. Charges are induced on the inner surface of the box, and an electric field appears inside the box, that is, outside the capacitor (Fig. 50).

Figure: 48. Condenser in a metal box

Figure: 49. Electric field of a charged flat capacitor

Figure: 50. Electric field of a charged capacitor placed in a metal box

But this means that charges will also appear on the outer surfaces of the capacitor plates. Since in this case the total charge of the insulated plate does not change, the charge on its outer surface can arise only due to the flow of charge from the inner surface. But when the charge changes on the inner surfaces of the plates, the field strength between the capacitor plates will change.

Thus, the enclosure of the considered capacitor in a metal box leads to a change in the electrical state of the internal space.

The change in plate charges and electric field in this example can be easily calculated. Let's denote the charge of an isolated capacitor through the Charge flowing to the outer surfaces of the plates when putting on the box, denote by The same charge of the opposite sign will be induced on the inner surfaces of the box. On the inner surfaces of the capacitor plates, a charge will remain.Then, in the space between the plates, the intensity of the uniform field will be equal in SI units, and outside the capacitor the field is directed in the opposite direction and its intensity is equal to where is the area of \u200b\u200bthe plate. Requiring that the potential difference between the opposite walls of the metal box be equal to zero, and assuming for simplicity the distances between all plates the same and equal then

This result is easy to understand if we take into account that after putting on the box, the field exists in all three gaps between the plates, i.e., in fact, there are three identical capacitors, the equivalent circuit of which is shown in Fig. 51. Calculating the capacity of the resulting system of capacitors, we get.

A metal box on the capacitor provides electrostatic protection to the system. Now we can bring any charged or uncharged bodies to the outside of the box without changing the electric field inside the box. This means that the capacity of the system will not change either.

Let's pay attention to the fact that in the analyzed example, having found out everything that interested us, we nevertheless bypassed the question of what forces carried out the redistribution of charges. What electric field caused the electrons to move in the conductive box material?

Obviously, this can only be that inhomogeneous field that goes beyond the capacitor near the edges of the plate (see Fig. 39). Although the strength of this field is small and is not taken into account when calculating the change in capacitance, it is it that determines the essence of the phenomenon under consideration - it moves the charges and thereby causes a change in the strength of the electric field inside the box.

Why the charge of a capacitor should not be understood as the full charge of the plate, but only that part of it that is on its inner side. facing the other cover?

What is the role of edge effects when considering electrostatic phenomena in a capacitor?

How will the capacity of a capacitor bank change if the plates of one of them are closed?

One of the most important parameters by which a capacitor is characterized is its electrical capacity (C). Physical quantity C, equal to:

called the capacitance of the capacitor. Where q is the magnitude of the charge of one of the capacitor plates, and is the potential difference between its plates. The capacitance of a capacitor is a value that depends on the size and design of the capacitor.

For capacitors with the same device and with equal charges on its plates, the potential difference of the air capacitor will be one times less than the potential difference between the plates of a capacitor, the space of which between the plates is filled with a dielectric with a dielectric constant. So the capacitance of a capacitor with a dielectric (C) is times greater than the electrical capacity of an air capacitor ():

where is the dielectric constant of the dielectric.

The unit of capacitance of a capacitor is the capacity of such a capacitor, which is charged by a unit charge (1 C) to a potential difference equal to one volt (in SI). The unit of capacitance of a capacitor (like any eclectic capacitance) in the International System of Units (SI) is the farad (F).

Electric capacity of a flat capacitor

In most cases, the field between the plates of a flat capacitor is considered uniform. The uniformity is broken only near the edges. When calculating the capacitance of a flat capacitor, these edge effects are usually neglected. This is possible if the distance between the plates is small in comparison with their linear dimensions. In this case, the capacitance of a flat capacitor is calculated as:

where is the electrical constant; S is the area of \u200b\u200beach (or the smallest) plate; d is the distance between the plates.

The electric capacitance of a flat capacitor, which contains N layers of dielectric, the thickness of each, the corresponding dielectric constant of the i-th layer, is equal to:

Electric capacity of a cylindrical capacitor

The design of a cylindrical capacitor includes two coaxial (coaxial) cylindrical conductive surfaces of different radii, the space between which is filled with a dielectric. The electric capacity of such a capacitor is found as:

where l is the height of the cylinders; - radius of the outer covering; - radius of the inner lining.

Capacities of a spherical capacitor

A spherical capacitor is called a capacitor, the plates of which are two concentric spherical conducting surfaces, the space between them is filled with a dielectric. The capacity of such a capacitor is found as:

where are the radii of the capacitor plates.

Examples of problem solving

EXAMPLE 1

The task

Plates of a flat air condenser carry a charge that is evenly distributed with surface density. In this case, the distance between its plates is equal. How much will the potential difference across the plates of this capacitor change if its plates are moved apart to a distance?

Decision

Let's make a drawing.

In the problem, when the distance between the capacitor plates changes, the charge on its plates does not change, the capacitance and potential difference on the plates change. The capacity of a flat air condenser is: