A solitary conductor is called, near which there are no other charged bodies, dielectrics, which could affect the distribution of charges of this conductor.
The ratio of the magnitude of the charge to the potential for a particular conductor is a constant value, called electrical capacity (capacity) With , .
Thus, the electric capacitance of a solitary conductor is numerically equal to the charge that must be imparted to the conductor in order to change its potential by one. Experience has shown that the electrical capacitance of a solitary conductor depends on its geometric dimensions, shape, dielectric properties of the environment and does not depend on the magnitude of the charge of the conductor.
Consider a solitary ball of radius R located in a homogeneous medium with permittivity e. Previously, it was obtained that the potential of the ball is equal to 
. Then the capacity of the ball 
, i.e. depends only on its radius.
The unit of capacitance is 1 farad (F). 1F is the capacitance of such a solitary conductor, the potential of which will change by 1V when a charge of 1C is imparted. Farad is a very large value, therefore, in practice, submultiple units are used: millifarad (mF, 1mF = 10 -3 F), microfarad (μF, 1μF = 10 -6 F), nanofarad (nF, 1nF = 10 -9 F), picofarad (pF, 1pF = 10 -12 F).
Solitary conductors, even very large sizes, have small capacitances. A solitary ball with a radius 1500 times greater than the radius of the Earth would have a capacity of 1F. The electrical capacitance of the Earth is 0.7 mF.
secluded a conductor is called, near which there are no other charged bodies, dielectrics, which could affect the distribution of charges of this conductor.
The ratio of the magnitude of the charge to the potential for a particular conductor is a constant value, called electrical capacity (capacity) With:
The capacitance of a solitary conductor is numerically equal to the charge that must be imparted to the conductor in order to change its potential by one. 1 farad (F) - 1 F is taken as a unit of capacitance.
Ball capacity = 4pεε 0 R.
Devices that have the ability to accumulate significant charges are called capacitors. A capacitor consists of two conductors separated by a dielectric. The electric field is concentrated between the plates, and the bound charges of the dielectric weaken it, i.e. lower the potential, which leads to a greater accumulation of charges on the capacitor plates. The capacitance of a flat capacitor is numerically equal to 
.
To vary the values of electrical capacity, capacitors are connected to batteries. In this case, their parallel and serial connections are used.
When capacitors are connected in parallel the potential difference on the plates of all capacitors is the same and equal to (φ A - φ B). The total charge of the capacitors is
Full battery capacity (fig.28) 
is equal to the sum of the capacitances of all capacitors; capacitors are connected in parallel when it is required to increase the capacitance and, therefore, the accumulated charge.
When capacitors are connected in series the total charge is equal to the charges of the individual capacitors 
, and the total potential difference is (Fig. 29)
, , 
.
From here.
When capacitors are connected in series, the reciprocal of the resulting capacitance is equal to the sum of the reciprocals of the capacitances of all capacitors. The resulting capacity is always less than the smallest capacity used in the battery.
The energy of a charged solitary conductor,
capacitor. Electrostatic field energy
The energy of a charged conductor is numerically equal to the work that external forces must do to charge it:
W= A. When transferring charge d q from infinity, work is done on the conductor d A against the forces of the electrostatic field (to overcome the Coulomb repulsive forces between like charges): d A= jd q= C jdj.
Consider a solitary conductor, i.e. e. a conductor that is removed from other conductors, bodies and charges. Its potential, according to, is directly proportional to the charge of the conductor. It follows from experience that different conductors, being equally charged, accept different potentials. Therefore, for a solitary conductor, we can write
the value
called electrical capacity(or simply capacity) solitary conductor. The capacitance of a solitary conductor is determined by the charge, the message of which to the conductor changes its potential by one.
The capacitance of the conductor depends on its size and shape, but does not depend on the material, state of aggregation, shape and size of the cavities inside the conductor. This is due to the fact that excess charges are distributed on the outer surface of the conductor. The capacitance also does not depend on the charge of the conductor, nor on its potential. The foregoing does not contradict the formula, since it only shows that the capacitance of a solitary conductor is directly proportional to its charge and inversely proportional to the potential.
Unit of electrical capacity - farad(F): 1F-capacity of such a solitary conductor, the potential of which changes to IB when an order of 1 C is given to it.
Accordingly, the potential of a solitary ball of radius R, located in a homogeneous medium with permittivity ε, is equal to
Using the formulas, we get that the capacity of the ball
It follows that a solitary ball in vacuum and having a radius R=C/(4π) 9-10 6 km, which is about 1400 times greater than the radius of the Earth, would have a capacity of 1 F (Earth's electric capacitance is C 0.7 mF). Consequently, the farad is a very large value, therefore, in practice, submultiple units are used - millifarad (mF), microfarad (μF), nanofarad (nF), picofarad (pF). It also follows from the formula that the unit of the electrical constant is farad per meter (F/m)
Capacitors
In order for a conductor to have a large capacitance, it must be very large. In practice, however, devices are needed that, with small sizes and small potentials relative to the surrounding bodies, can accumulate significant charges, in other words, have a large capacity. These devices are called capacitors.
A capacitor consists of two conductors (plates) separated by a dielectric. The capacitance of the capacitor should not be affected by the surrounding bodies, so the conductors are shaped so that the field created by the accumulated charges is concentrated in a narrow gap between the capacitor plates. This condition is satisfied: 1) two flat plates; 2) two coaxial cylinders; 3) two concentric spheres. Therefore, depending on the shape of the plates, capacitors are divided into flat, cylindrical and spherical.
Under capacitor capacity is understood as a physical quantity equal to the ratio of the charge Q accumulated in the capacitor to the potential difference ( - ) between its covers:
24. Connection of capacitors.
When connected in parallel capacitors, the battery charge is q=q1+q2, aU is the same and equal to the potential difference. The electrical capacity of the battery (C) is equal to C=C1+C2, with ncapacitors C=the sum of all electrical capacities.
When connected in series capacitors with electric capacitances C1 and C2, the total charge of the battery is equal to the charge of each capacitor (q=q1=q2). The total U is equal to the sum of the voltages on the individual capacitors: U=U1+U2. Electric capacity of the battery of two series capacitors: 1\C=1\C1+1\C2 or C=C1C2/(C1+C2). When connecting ncapacitors C=
25. The energy of the system of charges. The energy of a solitary charged conductor.
electrostatic interaction forces are conservative; This means that the system of charges has potential energy.
W1=Q1*ϕ12; W2=Q2*ϕ21
where φ 12 and φ 21 are, respectively, the potentials that are created by the charge Q 2 at the location of the charge Q 1 and the charge Q 1 at the location of the charge Q 2 . According to,
and
so W 1 = W 2 = W and
By adding to our system of two charges sequentially the charges Q 3 , Q 4 , ... , we can prove that in the case of n fixed charges, the interaction energy of the system of point charges is equal to
(1)
where φ i is the potential that is created at the point where the charge Q i is located, by all charges except the i-th.
The energy of a solitary charged conductor:
Consider a solitary conductor, the charge, potential and capacitance of which are respectively equal to Q, φ and C. Let us increase the charge of this conductor by dQ. To do this, it is necessary to transfer the charge dQ from infinity to a solitary conductor, while spending work on this, which is equal to
- elementary work of the forces of the electric field of a charged conductor "\u003e
To charge a body from zero potential to φ, work must be done
(2)
The energy of a charged conductor is equal to the work that must be done to charge this conductor:
(3)
Formula (3) can also be obtained and the conditions that the potential of the conductor at all its points is the same, since the surface of the conductor is equipotential. If φ is the potential of the conductor, we find 
where Q=∑Q i is the charge of the conductor.
26. The energy of a charged capacitor. The energy of the electrostatic field.
The capacitor consists of charged conductors, therefore, it has an energy, which from the formula is equal to
where Q is the charge of the capacitor, C is its capacitance, Δφ is the potential difference between the capacitor plates.
« Physics - Grade 10 "
Under what condition can a large electric charge be accumulated on conductors?
With any method of electrification of bodies - with the help of friction, an electrostatic machine, a galvanic cell, etc. - initially neutral bodies are charged due to the fact that some of the charged particles pass from one body to another.
Usually these particles are electrons.
When electrifying two conductors, for example, from an electrostatic machine, one of them acquires a charge of +q, and the other -q.
An electric field appears between the conductors and a potential difference (voltage) arises.
With an increase in the charge of the conductors, the electric field between them increases.
In a strong electric field (at a high voltage and, accordingly, at a high intensity), a dielectric (for example, air) becomes conductive.
The so-called breakdown dielectric: a spark jumps between the conductors, and they are discharged.
The less the voltage between the conductors increases with an increase in their charges, the more charge can be accumulated on them.
Electrical capacity.
We introduce a physical quantity that characterizes the ability of two conductors to accumulate an electric charge.
This value is called electrical capacity.
The voltage U between two conductors is proportional to the electric charges that are on the conductors (on one +|q|, and on the other -|q|).
Indeed, if the charges are doubled, then the electric field strength will become 2 times greater, therefore, the work done by the field when moving the charge will also increase 2 times, i.e., the voltage will increase 2 times.
Therefore, the ratio of the charge q of one of the conductors (there is a charge of the same modulus on the other) to the potential difference between this conductor and the neighboring one does not depend on the charge.
It is determined by the geometric dimensions of the conductors, their shape and mutual arrangement, as well as the electrical properties of the environment.
This allows us to introduce the concept of electric capacitance of two conductors.
The electrical capacity of two conductors is the ratio of the charge of one of the conductors to the potential difference between them:
The electric capacitance of a solitary conductor is equal to the ratio of the charge of the conductor to its potential, if all other conductors are at infinity and the potential of the infinitely distant point is zero.
The lower the voltage U between the conductors when they communicate charges +|q| and -|q|, the greater the electrical capacity of the conductors.
Large charges can be stored on conductors without causing breakdown of the dielectric.
But the electric capacity itself does not depend either on the charges communicated to the conductors, or on the voltage arising between them.
Units of electrical capacity.
Formula (14.22) allows you to enter a unit of electrical capacity.
The electric capacitance of two conductors is numerically equal to unity if, when giving them charges+1 C and-1 C there is a potential difference between them 1 V.
This unit is called farad(F); 1 F \u003d 1 C / V.
Due to the fact that the charge of 1 C is very large, the capacitance of 1 F is very large.
Therefore, in practice, fractions of this unit are often used: microfarad (μF) - 10 -6 F and picofarad (pF) - 10 -12 F.
An important characteristic of conductors is electrical capacity.
The electrical capacity of the conductors is the greater, the smaller the potential difference between them when they are given charges of opposite signs.
Capacitors.
You can find a system of conductors of a very large electrical capacity in any radio receiver or buy it in a store. It's called a capacitor. Now you will learn how such systems are arranged and what their electrical capacity depends on.
Systems of two conductors, called capacitors. A capacitor consists of two conductors separated by a dielectric layer, the thickness of which is small compared to the dimensions of the conductors. The conductors in this case are called facings capacitor.
The simplest flat capacitor consists of two identical parallel plates located at a small distance from each other (Fig. 14.33). 
If the charges of the plates are identical in absolute value and opposite in sign, then the electric field lines of force begin on the positively charged capacitor plate and end on the negatively charged one (Fig. 14.28). Therefore, almost the entire electric field concentrated inside the capacitor and uniformly. 
To charge a capacitor, you need to attach its plates to the poles of a voltage source, for example, to the poles of a battery. You can also connect the first plate to the pole of the battery, in which the other pole is grounded, and ground the second plate of the capacitor. Then on the grounded plate there will be a charge opposite in sign and equal in absolute value to the charge of the ungrounded plate. The charge of the same modulus will go into the ground.
Under capacitor charge understand the absolute value of the charge of one of the plates.
Capacitor capacitance is determined by formula (14.22).
The electric fields of surrounding bodies almost do not penetrate inside the capacitor and do not affect the potential difference between its plates. Therefore, the capacitance of a capacitor is practically independent of the presence of any other bodies near it.
Capacitance of a flat capacitor.
The geometry of a flat capacitor is completely determined by the area S of its plates and the distance d between them. The capacitance of a flat capacitor should depend on these values.
The larger the area of the plates, the more charge can be accumulated on them: q~S. On the other hand, the stress between the plates according to formula (14.21) is proportional to the distance d between them. Therefore, the capacity 
In addition, the capacitance of a capacitor depends on the properties of the dielectric between the plates. Since the dielectric weakens the field, the capacitance in the presence of the dielectric increases.
Let us test the dependencies obtained by us from the reasoning. To do this, take a capacitor, in which the distance between the plates can be changed, and an electrometer with a grounded case (Fig. 14.34). We connect the body and the rod of the electrometer with the capacitor plates with conductors and charge the capacitor. To do this, you need to touch the plate of the capacitor connected to the rod with an electrified stick. The electrometer will show the potential difference between the plates. 
Pushing the plates apart we find increase in potential difference. According to the definition of electric capacity (see formula (14.22)) this indicates its decrease. In accordance with dependence (14.23), the electrical capacity should indeed decrease with increasing distance between the plates.
By inserting a dielectric plate, such as organic glass, between the capacitor plates, we find reduction of potential difference. Hence, the capacitance of a flat capacitor in this case increases. The distance between the plates d can be very small, and the area S can be large. Therefore, with a small size, the capacitor can have a large electrical capacity.
For comparison: in the absence of a dielectric between the plates of a flat capacitor with an electrical capacity of 1 F and a distance between the plates d = 1 mm, it would have to have a plate area S = 100 km 2.
In addition, the capacitance of a capacitor depends on the properties of the dielectric between the plates. Since the dielectric weakens the field, the capacitance in the presence of the dielectric increases: where ε is the permittivity of the dielectric.
Series and parallel connection of capacitors. In practice, capacitors are often connected in various ways. Figure 14.40 shows serial connection three capacitors.
If points 1 and 2 are connected to a voltage source, then the charge +qy will pass to the left plate of the capacitor C1, and the charge -q will pass to the right plate of the capacitor C3. Due to electrostatic induction, the right plate of capacitor C1 will have a charge of -q, and since the plates of capacitors C1 and C2 are connected and were electrically neutral before the voltage was connected, then, according to the law of conservation of charge, a charge of + q will appear on the left plate of capacitor C2, etc. On all plates of capacitors with such a connection will have the same charge in absolute value:
q \u003d q 1 \u003d q 2 \u003d q 3.
To determine the equivalent electrical capacity means to determine the electrical capacity of such a capacitor, which, with the same potential difference, will accumulate the same charge q as the system of capacitors.
The potential difference φ1 - φ2 is the sum of the potential differences between the plates of each of the capacitors:
φ 1 - φ 2 \u003d (φ 1 - φ A) + (φ A - φ B) + (φ B - φ 2),
or U \u003d U 1 + U 2 + U 3.
Using formula (14.23), we write:
Figure 14 41 shows a diagram connected in parallel capacitors. The potential difference between the plates of all capacitors is the same and equal to:
φ 1 - φ 2 \u003d U \u003d U 1 \u003d U 2 \u003d U 3.
Charges on the plates of capacitors
q 1 = C 1 U, q 2 = C 2 U, q 3 = C 3 U.
On an equivalent capacitor with a capacity of C eq, the charge on the plates at the same potential difference
q \u003d q 1 + q 2 + q 3.
For electrical capacity, according to formula (14.23), we write: C eq U \u003d C 1 U + C 2 U + C 3 U, therefore, C eq \u003d C 1 + C 2 + C 3, and in the general case
Various types of capacitors.
Capacitors have different devices depending on their purpose. A conventional technical paper capacitor consists of two strips of aluminum foil insulated from each other and from the metal case by paraffin-impregnated paper strips. The strips and ribbons are tightly folded into a small package.
In radio engineering, capacitors of variable electrical capacity are widely used (Fig. 14.35). Such a capacitor consists of two systems of metal plates, which, when the handle is rotated, can enter one into the other. In this case, the areas of overlapping parts of the plates and, consequently, their electric capacitance change. The dielectric in these capacitors is air. 
A significant increase in electrical capacity due to a decrease in the distance between the plates is achieved in the so-called electrolytic capacitors (Fig. 14.36). The dielectric in them is a very thin film of oxides covering one of the plates (foil strip). Another lining is paper impregnated with a solution of a special substance (electrolyte). 
Capacitors allow you to store electrical charge. The capacitance of a flat capacitor is proportional to the area of the plates and inversely proportional to the distance between the plates. In addition, it depends on the properties of the dielectric between the plates.
secluded called a conductor located so far from other bodies that the influence of the charges and fields of other bodies can be neglected. When a certain charge is imparted to such a conductor, it will be located on its surface in some way so that the equilibrium conditions are satisfied. In the surrounding space, the charge of the conductor will create an electric field. If an infinitely small (not affecting the charge of the conductor) charge is moved from the surface of the conductor to an infinitely small distance, then the field forces will do some work. The ratio gives the potential of the conductor, which he acquired as a result of giving him a charge.
If the conductor is additionally informed of the charge of one more portion of the charge, then it will be distributed over the surface in the same way as the first portion. Accordingly, at all points in space, the electric field strength will double. The work will also increase, and hence the potential of the conductor. Thus, it turns out that the charge imparted to the conductor and the potential acquired by it proportional . Therefore, we can write the ratio:
|
Proportionality factor With in relation (16.3) characterizes the ability of the conductor to accumulate an electric charge and is called the electrical capacity of a solitary conductor. This explorer option measured in farads . An electric capacitance of 1 farad is possessed by a conductor, which, when a charge of 1 coulomb is imparted, acquires a potential of 1 volt.
We calculate the capacitance of a solitary spherical conductor located in a medium with a dielectric constant. The field strength of a charged sphere outside its limits is described by an expression similar to the expression for the field strength of a point charge located in the center of the sphere. Therefore, the expression for the work of moving a small point charge from the surface of a sphere of radius , having a charge , to infinity has the form:
So capacitance of a solitary sphere is defined by the expression:
|
Substituting in (16.6) the radius of the Earth , we get the electrical capacity of the Earth, which is approximately 700 μF.
Capacitors
Solitary conductors have little capacitance. However, in technology, devices are used that have an electrical capacity of up to several farads. Such devices are capacitors . The principle of the device of capacitors is based on the fact that when approaching a solitary charged conductor of another (even uncharged) conductor, the electrical capacity of the system increases significantly. In the field of a solitary conductor, induced charges arise on the approaching body, and the charges of the sign opposite to the communicated solitary conductor are located closer to it and affect its field more strongly. The potential of the conductor modulo decreases, and the charge is conserved. It means that its electrical capacity increases.
The remote parts of the approaching conductor can be connected to the Earth (grounded), so that the induced charge of the same sign as that imparted to the solitary conductor is distributed over the surface of the Earth and does not affect the potential of the system. Obviously, by bringing the oppositely charged conductors as close as possible, a noticeable increase in electrical capacity can be achieved. Accordingly, capacitors are made flat when oppositely charged conductors ( capacitor plates ) in the form of, for example, foil strips, separated by a thin dielectric layer. In this case, the electric field of the system is concentrated in the space between the plates, and external bodies do not affect the capacitance of the capacitor. You can also imagine the plates in the form of concentric cylinders or spheres.
Capacitor capacitance, by definition, is the value of the ratio of the charge of each of the plates to the potential difference between them:
.The dielectric constant of the material between the plates of the capacitor.
