The work of the field on the movement of a charge is formula. Work on moving a charge in an electric field. Potential. Electric field formation and its features

For every charge in an electric field there is a force that can move this charge. Determine the work A of moving a point positive charge q from point O to point n, performed by the forces of the electric field of a negative charge Q. According to Coulomb’s law, the force moving the charge is variable and equal to

Where r is the variable distance between charges.

. This expression can be obtained like this:

The quantity represents the potential energy W p of the charge at a given point in the electric field:

The sign (-) shows that when a charge is moved by a field, its potential energy decreases, turning into the work of movement.

A value equal to the potential energy of a unit positive charge (q = +1) is called the electric field potential.

Then . For q = +1.

Thus, the potential difference between two points of the field is equal to the work of the field forces to move a unit positive charge from one point to another.

The potential of an electric field point is equal to the work done to move a unit positive charge from a given point to infinity: . Unit of measurement - Volt = J/C.

The work of moving a charge in an electric field does not depend on the shape of the path, but depends only on the potential difference between the starting and ending points of the path.

A surface at all points of which the potential is the same is called equipotential.

The field strength is its power characteristic, and the potential is its energy characteristic.

The relationship between field strength and its potential is expressed by the formula

,

the sign (-) is due to the fact that the field strength is directed in the direction of decreasing potential, and in the direction of increasing potential.

5. Use of electric fields in medicine.

Franklinization, or “electrostatic shower”, is a therapeutic method in which the patient’s body or certain parts of it are exposed to a constant high-voltage electric field.

The constant electric field during the general exposure procedure can reach 50 kV, with local exposure 15 - 20 kV.

Mechanism of therapeutic action. The franklinization procedure is carried out in such a way that the patient’s head or another part of the body becomes like one of the capacitor plates, while the second is an electrode suspended above the head or installed above the site of exposure at a distance of 6 - 10 cm. Under the influence of high voltage under the tips of the needles attached to the electrode, air ionization occurs with the formation of air ions, ozone and nitrogen oxides.

Inhalation of ozone and air ions causes a reaction in the vascular network. After a short-term spasm of blood vessels, capillaries expand not only in superficial tissues, but also in deep ones. As a result, metabolic and trophic processes are improved, and in the presence of tissue damage, the processes of regeneration and restoration of functions are stimulated.

As a result of improved blood circulation, normalization of metabolic processes and nerve function, there is a decrease in headaches, high blood pressure, increased vascular tone, and a decrease in pulse.

The use of franklinization is indicated for functional disorders of the nervous system

Examples of problem solving

1. When the franklinization apparatus operates, 500,000 light air ions are formed every second in 1 cm 3 of air. Determine the work of ionization required to create the same amount of air ions in 225 cm 3 of air during a treatment session (15 min). The ionization potential of air molecules is assumed to be 13.54 V, and air is conventionally considered to be a homogeneous gas.

- ionization potential, A - ionization work, N - number of electrons.

2. When treating with an electrostatic shower, a potential difference of 100 kV is applied to the electrodes of the electric machine. Determine how much charge passes between the electrodes during one treatment procedure, if it is known that the electric field forces do 1800 J of work.

From here

Electric dipole in medicine

In accordance with Einthoven's theory, which underlies electrocardiography, the heart is an electric dipole located in the center of an equilateral triangle (Einthoven triangle), the vertices of which can conventionally be considered

located in the right hand, left hand and left leg.

During the cardiac cycle, both the position of the dipole in space and the dipole moment change. Measuring the potential difference between the vertices of the Einthoven triangle allows us to determine the relationship between the projections of the dipole moment of the heart onto the sides of the triangle as follows:

Knowing the voltages U AB, U BC, U AC, you can determine how the dipole is oriented relative to the sides of the triangle.

In electrocardiography, the potential difference between two points on the body (in this case, between the vertices of Einthoven's triangle) is called a lead.

Registration of the potential difference in leads depending on time is called electrocardiogram.

The geometric location of the end points of the dipole moment vector during the cardiac cycle is called vector cardiogram.

Lecture No. 4

Contact phenomena

1. Contact potential difference. Volta's laws.

2. Thermoelectricity.

3. Thermocouple, its use in medicine.

4. Resting potential. Action potential and its distribution.

1. Contact potential difference. Volta's laws.

When dissimilar metals come into close contact, a potential difference arises between them, depending only on their chemical composition and temperature (Volta's first law). This potential difference is called contact.

In order to leave the metal and go into the environment, the electron must do work against the forces of attraction towards the metal. This work is called the work function of an electron leaving the metal.

Let us bring into contact two different metals 1 and 2, having work function A 1 and A 2, respectively, and A 1< A 2 . Очевидно, что свободный электрон, попавший в процессе теплового движения на поверхность раздела металлов, будет втянут во второй металл, так как со стороны этого металла на электрон действует большая сила притяжения (A 2 >A 1). Consequently, through the contact of metals, free electrons are “pumped” from the first metal to the second, as a result of which the first metal is charged positively, the second - negatively. The potential difference that arises in this case creates an electric field of intensity E, which makes it difficult for further “pumping” of electrons and will completely stop when the work of moving an electron due to the contact potential difference becomes equal to the difference in the work functions:

(1)

Let us now bring into contact two metals with A 1 = A 2, having different concentrations of free electrons n 01 > n 02. Then the preferential transfer of free electrons from the first metal to the second will begin. As a result, the first metal will be charged positively, the second - negatively. A potential difference will arise between the metals, which will stop further electron transfer. The resulting potential difference is determined by the expression:

, (2)

where k is Boltzmann's constant.

In the general case of contact between metals that differ in both the work function and the concentration of free electrons, the cr.r.p. from (1) and (2) will be equal to:

(3)

It is easy to show that the sum of the contact potential differences of series-connected conductors is equal to the contact potential difference created by the end conductors and does not depend on the intermediate conductors:

This position is called Volta's second law.

If we now directly connect the end conductors, then the potential difference existing between them is compensated by an equal potential difference that arises in contact 1 and 4. Therefore, the c.r.p. does not create current in a closed circuit of metal conductors having the same temperature.

2. Thermoelectricity is the dependence of the contact potential difference on temperature.

Let's make a closed circuit of two dissimilar metal conductors 1 and 2.

The temperatures of contacts a and b will be maintained at different temperatures T a > T b . Then, according to formula (3), c.r.p. in the hot junction more than in the cold junction: . As a result, a potential difference arises between junctions a and b, called thermoelectromotive force, and current I will flow in the closed circuit. Using formula (3), we obtain

Where for each pair of metals.

1. Thermocouple, its use in medicine.

A closed circuit of conductors that creates current due to differences in contact temperatures between the conductors is called thermocouple.

From formula (4) it follows that the thermoelectromotive force of a thermocouple is proportional to the temperature difference of the junctions (contacts).

Formula (4) is also valid for temperatures on the Celsius scale:

A thermocouple can only measure temperature differences. Typically one junction is maintained at 0ºC. It's called the cold junction. The other junction is called the hot or measuring junction.

The thermocouple has significant advantages over mercury thermometers: it is sensitive, inertia-free, allows you to measure the temperature of small objects, and allows remote measurements.

Measuring the temperature field profile of the human body.

It is believed that the human body temperature is constant, but this constancy is relative, since in different parts of the body the temperature is not the same and varies depending on the functional state of the body.

Skin temperature has its own well-defined topography. The lowest temperature (23-30º) is found in the distal limbs, tip of the nose, and ears. The highest temperature is in the armpits, perineum, neck, lips, cheeks. The remaining areas have a temperature of 31 - 33.5 ºС.

In a healthy person, the temperature distribution is symmetrical relative to the midline of the body. Violation of this symmetry serves as the main criterion for diagnosing diseases by constructing a temperature field profile using contact devices: a thermocouple and a resistance thermometer.

4. Resting potential. Action potential and its distribution.

The surface membrane of a cell is not equally permeable to different ions. In addition, the concentration of any specific ions is different on different sides of the membrane; the most favorable composition of ions is maintained inside the cell. These factors lead to the appearance in a normally functioning cell of a potential difference between the cytoplasm and the environment (resting potential)

When excited, the potential difference between the cell and the environment changes, an action potential arises, which propagates in the nerve fibers.

The mechanism of action potential propagation along a nerve fiber is considered by analogy with the propagation of an electromagnetic wave along a two-wire line. However, along with this analogy, there are also fundamental differences.

An electromagnetic wave, propagating in a medium, weakens as its energy dissipates, turning into the energy of molecular-thermal motion. The source of energy of an electromagnetic wave is its source: generator, spark, etc.

The excitation wave does not decay, since it receives energy from the very medium in which it propagates (the energy of the charged membrane).

Thus, the propagation of an action potential along a nerve fiber occurs in the form of an autowave. The active environment is excitable cells.

Examples of problem solving

1. When constructing a profile of the temperature field of the surface of the human body, a thermocouple with a resistance of r 1 = 4 Ohms and a galvanometer with a resistance of r 2 = 80 Ohms are used; I=26 µA at a junction temperature difference of ºС. What is the thermocouple constant?

The thermopower arising in a thermocouple is equal to , where thermocouples is the temperature difference between the junctions.

According to Ohm's law, for a section of the circuit where U is taken as . Then

Lecture No. 5

Electromagnetism

1. The nature of magnetism.

2. Magnetic interaction of currents in a vacuum. Ampere's law.

4. Dia-, para- and ferromagnetic substances. Magnetic permeability and magnetic induction.

5. Magnetic properties of body tissues.

1. The nature of magnetism.

A magnetic field arises around moving electric charges (currents), through which these charges interact with magnetic or other moving electric charges.

A magnetic field is a force field and is represented by magnetic lines of force. Unlike electric field lines, magnetic field lines are always closed.

The magnetic properties of a substance are caused by elementary circular currents in the atoms and molecules of this substance.

2 . Magnetic interaction of currents in a vacuum. Ampere's law.

The magnetic interaction of currents was studied using moving wire circuits. Ampere established that the magnitude of the force of interaction between two small sections of conductors 1 and 2 with currents is proportional to the lengths of these sections, the current strengths I 1 and I 2 in them and is inversely proportional to the square of the distance r between the sections:

It turned out that the force of influence of the first section on the second depends on their relative position and is proportional to the sines of the angles and .

where is the angle between and the radius vector r 12 connecting with, and is the angle between and the normal n to the plane Q containing the section and the radius vector r 12.

Combining (1) and (2) and introducing the proportionality coefficient k, we obtain the mathematical expression of Ampere’s law:

(3)

The direction of the force is also determined by the gimlet rule: it coincides with the direction of translational movement of the gimlet, the handle of which rotates from normal n 1.

A current element is a vector equal in magnitude to the product Idl of an infinitely small section of length dl of a conductor and the current strength I in it and directed along this current. Then, passing in (3) from small to infinitesimal dl, we can write Ampere’s law in differential form:

. (4)

The coefficient k can be represented as

where is the magnetic constant (or magnetic permeability of vacuum).

The value for rationalization taking into account (5) and (4) will be written in the form

. (6)

3 . Magnetic field strength. Ampere's formula. Biot-Savart-Laplace Law.

Since electric currents interact with each other through their magnetic fields, a quantitative characteristic of the magnetic field can be established on the basis of this interaction - Ampere's law. To do this, we divide the conductor l with current I into many elementary sections dl. It creates a field in space.

At point O of this field, located at a distance r from dl, we place I 0 dl 0. Then, according to Ampere’s law (6), a force will act on this element

(7)

where is the angle between the direction of current I in the section dl (creating the field) and the direction of the radius vector r, and is the angle between the direction of current I 0 dl 0 and the normal n to the plane Q containing dl and r.

In formula (7) we select the part that does not depend on the current element I 0 dl 0, denoting it by dH:

Biot-Savart-Laplace law (8)

The value of dH depends only on the current element Idl, which creates a magnetic field, and on the position of point O.

The value dH is a quantitative characteristic of the magnetic field and is called magnetic field strength. Substituting (8) into (7), we get

where is the angle between the direction of the current I 0 and the magnetic field dH. Formula (9) is called the Ampere formula and expresses the dependence of the force with which the magnetic field acts on the current element I 0 dl 0 located in it on the strength of this field. This force is located in the Q plane perpendicular to dl 0. Its direction is determined by the “left hand rule”.

Assuming =90º in (9), we get:

Those. The magnetic field strength is directed tangentially to the field line and is equal in magnitude to the ratio of the force with which the field acts on a unit current element to the magnetic constant.

4 . Diamagnetic, paramagnetic and ferromagnetic substances. Magnetic permeability and magnetic induction.

All substances placed in a magnetic field acquire magnetic properties, i.e. are magnetized and therefore change the external field. In this case, some substances weaken the external field, while others strengthen it. The first ones are called diamagnetic, second – paramagnetic substances. Among paramagnetic substances, a group of substances stands out sharply, causing a very large increase in the external field. This ferromagnets.

Diamagnets- phosphorus, sulfur, gold, silver, copper, water, organic compounds.

Paramagnets- oxygen, nitrogen, aluminum, tungsten, platinum, alkali and alkaline earth metals.

Ferromagnets– iron, nickel, cobalt, their alloys.

The geometric sum of the orbital and spin magnetic moments of electrons and the intrinsic magnetic moment of the nucleus forms the magnetic moment of an atom (molecule) of a substance.

In diamagnetic materials, the total magnetic moment of an atom (molecule) is zero, because magnetic moments cancel each other out. However, under the influence of an external magnetic field, a magnetic moment is induced in these atoms, directed opposite to the external field. As a result, the diamagnetic medium becomes magnetized and creates its own magnetic field, directed opposite to the external one and weakening it.

The induced magnetic moments of diamagnetic atoms are preserved as long as an external magnetic field exists. When the external field is eliminated, the induced magnetic moments of the atoms disappear and the diamagnetic material is demagnetized.

In paramagnetic atoms, the orbital, spin, and nuclear moments do not compensate each other. However, atomic magnetic moments are arranged randomly, so the paramagnetic medium does not exhibit magnetic properties. An external field rotates the paramagnetic atoms so that their magnetic moments are established predominantly in the direction of the field. As a result, the paramagnetic material becomes magnetized and creates its own magnetic field, coinciding with the external one and enhancing it.

(4), where is the absolute magnetic permeability of the medium. In vacuum =1, , and

In ferromagnets there are regions (~10 -2 cm) with identically oriented magnetic moments of their atoms. However, the orientation of the domains themselves is varied. Therefore, in the absence of an external magnetic field, the ferromagnet is not magnetized.

With the appearance of an external field, domains oriented in the direction of this field begin to increase in volume due to neighboring domains having different orientations of the magnetic moment; the ferromagnet becomes magnetized. With a sufficiently strong field, all domains are reoriented along the field, and the ferromagnet is quickly magnetized to saturation.

When the external field is eliminated, the ferromagnet is not completely demagnetized, but retains residual magnetic induction, since thermal motion cannot disorient the domains. Demagnetization can be achieved by heating, shaking or applying a reverse field.

At a temperature equal to the Curie point, thermal motion is capable of disorienting atoms in domains, as a result of which the ferromagnet turns into a paramagnet.

The flux of magnetic induction through a certain surface S is equal to the number of induction lines penetrating this surface:

(5)

Unit of measurement B – Tesla, F-Weber.

When a charge moves in an electrostatic field, the Coulomb forces acting on the charge do work. Let charge q 0 0 move in the field of charge q0 from point C to point B along an arbitrary trajectory (Fig. 1.12). The Coulomb force acts on q 0

With elementary charge movement d l, this force does work dA

Where  is the angle between the vectors and. Value d l cos=dr is the projection of the vector onto the direction of the force. Thus, dA=Fdr, . The total work done to move a charge from point C to B is determined by the integral , where r 1 and r 2 are the distances of the charge q to points C and B. From the resulting formula it follows that the work done when moving an electric charge q 0 in the field of a point charge q, does not depend on the shape of the movement path, but depends only on the starting and ending points of movement .

In the dynamics section it is shown that a field that satisfies this condition is potential. Therefore, the electrostatic field of a point charge is potential, and the forces acting in it are conservative.

If the charges q and q 0 are of the same sign, then the work of the repulsive forces will be positive when they move away and negative when they approach (in the latter case, the work is performed by external forces). If the charges q and q 0 are opposite, then the work of the attractive forces will be positive when they approach each other and negative when they move away from each other (in the latter case, the work is also performed by external forces).

Let the electrostatic field in which the charge q 0 moves be created by a system of charges q 1, q 2,...,q n. Consequently, independent forces act on q 0 , whose resultant is equal to their vector sum. The work A of the resultant force is equal to the algebraic sum of the work of the component forces, , where r i1 and r i2 are the initial and final distances between charges q i and q 0.

Circulation of the tension vector.

When a charge moves along an arbitrary closed path L, the work done by the electrostatic field forces is zero. Since the final position of the charge is equal to the initial position r 1 =r 2, then (the circle near the integral sign indicates that the integration is carried out along a closed path). Since and , then . From here we get . Reducing both sides of the equality by q 0, we obtain or, where E l=Ecos - projection of vector E onto the direction of elementary displacement. The integral is called circulation of the tension vector. Thus, circulation of the electrostatic field strength vector along any closed loop is zero . This conclusion is a condition field potentiality.

Potential charge energy.

In a potential field, bodies have potential energy and the work of conservative forces is done due to the loss of potential energy.

Therefore work A 12 can be represented as the difference in potential charge energies q 0 at the initial and final points of the charge field q :

Potential charge energy q 0 located in the charge field q on distance r equal to

Assuming that when the charge is removed to infinity, the potential energy goes to zero, we get: const = 0 .

For namesake charges potential energy of their interaction ( repulsion) positive, For different names charges potential energy from interaction ( attraction) negative.

If the field is created by the system n point charges, then the potential energy of the charge q 0 located in this field is equal to the sum of its potential energies created by each of the charges separately:

Electrostatic field potential.

The ratio does not depend on the test charge q0 and is, energy characteristic of the field, called potential :

Potential ϕ at any point in the electrostatic field is scalar physical quantity, determined by the potential energy of a unit positive charge placed at this point.

Potential electrostatic field - a scalar quantity equal to the ratio of the potential energy of a charge in the field to this charge: Energy characteristics of the field at a given point. The potential does not depend on the amount of charge placed in this field.
Because potential energy depends on the choice of coordinate system, then the potential is determined accurate to a constant. The reference point for the potential is chosen depending on the task: a) the potential of the Earth, b) the potential of an infinitely distant point of the field, c) the potential of the negative plate of the capacitor.
A consequence of the principle of field superposition (potentials add up algebraically).
The potential is numerically equal to the work of the field in moving a unit positive charge from a given point of the electric field to infinity. In SI, potential is measured in volts:
Potential difference
Voltage - the difference in potential values ​​at the initial and final points of the trajectory. Voltage is numerically equal to the work of the electrostatic field when a unit positive charge moves along the lines of force of this field. The potential difference (voltage) is independent of the selection coordinate systems!
Unit of potential difference The voltage is 1 V if, when moving a positive charge of 1 C along the lines of force, the field does 1 J of work.
Connection between tension and tension.
From what was proven above: → the tension is equal to the potential gradient (the rate of change of potential along the direction d).
From this ratio it is clear:
Equipotential surfaces. EPP - surfaces of equal potential. EPP properties: No work is done when moving a charge along an equipotential surface; The tension vector is perpendicular to the EPP at each point.
Electrical voltage (potential difference) measurement There is an electric field between the rod and the body. Measuring the potential of a conductor Measuring the voltage across a galvanic cell An electrometer is more accurate than a voltmeter.

A system of charged bodies has potential energy, called electrostatic, because An electrostatic field can move charged bodies placed in it, while doing work.

Let us consider the work of electrostatic forces to move a charge q in a uniform electrostatic field with intensity E, created by two infinitely large plates with charges equal in magnitude and opposite in sign. Let us associate the origin of the coordinate axis with the negatively charged plate. A point charge q in a field is acted upon by a force. When a charge moves from point 1 to point 2 along a power line, the electrostatic field does work .

When moving a charge from point 1 to point 3. But . Hence, .

The work of electrostatic forces when moving an electric charge from point 1 to point 3 is calculated according to the derived formula for any trajectory shape. If a charge moves along a curve, then it can be divided into very small straight sections along the field strength and perpendicular to it. No work is done in areas perpendicular to the field. The sum of the projections of the remaining sections onto the power line is equal to d 1 -d 2, i.e.

.

Thus, the work done when moving a charge in a uniform electrostatic field does not depend on the shape of the trajectory along which the charge moves, but depends only on the coordinates of the starting and ending points of the path. This conclusion is also valid for a non-uniform electrostatic field. Consequently, the Coulomb force is potential or conservative and its work when moving charges is associated with a change in potential energy. The work of conservative forces does not depend on the shape of the body's trajectory and is equal to the change in the potential energy of the body, taken with the opposite sign.

.

. Means, .

It is not the potential energy itself that has an exact physical meaning, because its numerical value depends on the choice of the origin, and the change in potential energy, because only it is determined unambiguously.

The work of the electrostatic field when moving a charge along a closed path is zero, because d 2 =d 1.

A QUALITY EQUAL TO THE POTENTIAL ENERGY PER UNIT POSITIVE CHARGE PLACED AT A GIVEN POINT OF THE ELECTROSTATIC FIELD IS CALLED THE POTENTIAL OF THE ELECTROSTATIC FIELD AT A GIVEN POINT.

Potential is a scalar quantity. This is the energy characteristic of the field, because determines the potential energy of the charge at a given point.

The potential is determined up to a certain constant, the value of which depends on the choice of the zero level of potential energy. As the charge creating the field moves away in a non-uniform field, the field weakens. This means that its potential also decreases.j = O at a point infinitely distant from the charge. Consequently, the field potential at a given point in the field is the work done by electrostatic forces when moving a unit positive charge from this point to an infinitely distant one. The potential of any point in the field created by a positive charge is positive. In electrical engineering, the surface of the Earth is taken to be a surface with zero potential.

Potential difference - the difference in potential values ​​at the initial and final points of the trajectory.

.

The potential difference between two points is the work done by Coulomb forces to move a unit positive charge between them. The potential difference has a precise physical meaning, because does not depend on the choice of reference system.

[V]=J/Cl=V. 1 volt is the potential difference between points, when moving between which a charge of 1 C, Coulomb forces do 1 J of work.

Let us calculate the potential of the points of the field created by the point charge Q.

Let charge q move in the field of charge Q along a radial straight line. The charge moves in a non-uniform field. Consequently, when moving, the force acting on the charge will change. But you can divide the entire movement into small sections dr, on each of which the force can be considered constant. Then, . Then work all the way

Work in an electrostatic field does not depend on the shape of the trajectory.

Therefore, if the charge moves from the charge creating the field, not along a radial straight line, then it can be moved from the initial point to the final point by moving it first along a circular arc of radius r 1, and then along a radial segment to the end point. No work will be done in the first section, because... the Coulomb force will be perpendicular to the speed of the body, and on the second it will be found according to the formula found above.

The potential of the resulting field of a system of charges at a given point, according to the principle of field superposition, is equal to the algebraic sum of the potentials of the component fields at this point.

The geometric locus of points in a field of equal potential is called an EQUIPOTENTIAL SURFACE. Equipotential surfaces are perpendicular to the lines of force. The work done by the field when a charge moves along an equipotential surface is zero. The surface of a conductor in an electrostatic field is equipotential. The potential of all points inside a conductor is equal to the potential on its surface. Otherwise, there would be a potential difference between the points of the conductor, which would lead to the generation of electric current. Equipotential surfaces cannot intersect.

Unlike other quantities in electrostatics, the potential difference between bodies can be easily measured using an electrometer by connecting the body and its arrow to the bodies located at these points. In this case, the angle of deflection of the electrometer needle is determined only by the potential difference between the bodies (or, what is the same, between the needle and the body of the electrometer). In practice, the potential difference between points in electrical circuits is measured by a voltmeter connected to these points.

The work done to move an electric charge in a uniform electrostatic field can be found through the force characteristic of the field - tension, and through the energy characteristic - potential. This allows you to establish a connection between them.

Hence:

This relationship allows us to introduce the SI unit of field strength. . The intensity of a uniform electrostatic field is equal to if the potential difference between points lying on the same field line at a distance of 1 m is equal to 1 V.

In an electrostatic field, the tension is directed in the direction of decreasing potential.

It is easy to show that in inhomogeneous fields:

The “-” sign indicates that the potential decreases along the field line.

When moving from one medium to another, potential, unlike tension, cannot change abruptly.

ELECTRIC CAPACITY.

The potential of an isolated conductor is proportional to the charge imparted to it. The ratio of the charge on a conductor to its potential does not depend on the amount of charge. It characterizes the ability of a given conductor to accumulate charges on itself. THE ELECTRIC CAPACITY OF A SOLE CONDUCTOR IS A VALUE EQUAL TO THE ELECTRIC CHARGE THAT CHANGES THE POTENTIAL OF THE CONDUCTOR BY UNIT . To calculate the electrical capacity of an isolated conductor, it is necessary to divide the charge imparted to it by the potential that arises on it.

1 farad is the electrical capacity of a conductor, the potential of which changes by 1 V when a charge of 1 C is imparted to it. A farad is a huge capacitance, so in practice we deal with micro- and picofarads. The electrical capacity of a conductor depends on its geometric dimensions, shape and dielectric constant of the medium in which it is located, as well as on the location of surrounding bodies.

Ball potential. Therefore, its electrical capacity

When a charge is transferred from one of the uncharged conductors to another, a potential difference arises between them, proportional to the amount of the transferred charge. The ratio of the module of the transferred charge to the resulting potential difference does not depend on the magnitude of the transferred charge. It characterizes the ability of these two bodies to accumulate an electric charge. THE MUTUAL ELECTRIC CAPACITY OF TWO CONDUCTORS IS A QUALITY EQUAL TO THE CHARGE THAT MUST BE TRANSFERRED FROM ONE CONDUCTOR TO ANOTHER TO CHANGE THE POTENTIAL DIFFERENCE BETWEEN THEM BY UNIT.

The mutual electrical capacitance of bodies depends on the size and shape of the bodies, on the distance between them, on the dielectric constant of the medium in which they are located.

They have high electrical capacity capacitors - a system of two or more conductors, called plates, separated by a layer of dielectric . The charge of a capacitor is the charge modulus of one of the plates.

To charge a capacitor, its plates are connected to the poles of a current source or, having grounded one of the plates, the second is connected to any pole of the source, the second pole of which is also grounded.

The electrical capacity of a capacitor is the charge whose message to the capacitor causes the appearance of a unit potential difference between the plates. To calculate the electrical capacity of a capacitor, you need to divide its charge by the potential difference between the plates.

Let the distance between the plates of a flat capacitor d be much smaller than their dimensions. Then the field between the plates can be considered uniform, and the plates can be considered infinite charged planes. Electrostatic field strength from one plate: . General tension:

Potential difference between plates:

. =>

This formula is valid for small d, i.e. with a uniform field inside the capacitor.

There are capacitors of constant, variable and semi-variable capacitance (trimmers). Constant capacitors are usually named after the type of dielectric between the plates: mica, ceramic, paper.

In variable capacitors, the dependence of the capacitance on the overlap area of ​​the plates is often used.

For trimmers (or tuning capacitors), the capacitance changes when tuning radio devices, but remains constant during operation.

§ 12.3 Work of electrostatic field forces. Potential. Equipotential surfaces

A charge q pr placed at an arbitrary point of an electrostatic field with intensity E is acted upon by a force F = q pr E. If the charge is not fixed, then the force will make it move and, therefore, work will be done. The elementary work done by force F when moving a point electric charge q pr from point a of the electric field to point b on the path segment dℓ, by definition, is equal to

(α is the angle between F and the direction of movement) (Fig. 12.13).

If work is done by external forces, then dA< 0 , если силами поля, то dA >0. Integrating the last expression, we obtain that the work against field forces when moving q pr from the point a to point b

(12.20)

Figure -12.13

(
- Coulomb force acting on the test charge q pr at each point of the field with intensity E).

Then work

(12.21)

The movement occurs perpendicular to the vector , therefore cosα =1, work of transfer of test charge q from a To b equal to

(12.22)

The work of electric field forces when moving a charge does not depend on the shape of the path, but depends only on the relative position of the starting and ending points of the trajectory.

Therefore, the electrostatic field of a point charge ispotential , and electrostatic forces –conservative .

This is a property of potential fields. It follows from it that the work done in an electric field along a closed circuit is equal to zero:

(12.23)

Integral
called circulation of the tension vector . From the vanishing of the circulation of vector E, it follows that the lines of electric field strength cannot be closed; they begin on positive charges and end on negative charges.

As is known, the work of conservative forces is accomplished due to the loss of potential energy. Therefore, the work of electrostatic field forces can be represented as the difference in potential energies possessed by a point charge q at the initial and final points of the field of charge q:

(12.24)

whence it follows that the potential energy of charge q in the field of charge q is equal to

(12.25)

For like charges q pr q >0 and the potential energy of their interaction (repulsion) is positive, for unlike charges q pr q< 0 и потенциальная энергия их взаимодействия (притяжения) отрицательна.

If the field is created by a system of n point charges q 1, q 2, …. q n, then the potential energy U of charge q pr located in this field is equal to the sum of its potential energies U i created by each of the charges separately:

(12.26)

Attitude do not depend on charge q and is an energy characteristic of the electrostatic field.

A scalar physical quantity measured by the ratio of the potential energy of a test charge in an electrostatic field to the magnitude of this charge is calledelectrostatic field potential.

(12.27)

The field potential created by a point charge q is equal to

(12.28)

Unit of potential – volt.

The work done by the forces of the electrostatic field when moving a charge q pr from point 1 to point 2 can be represented as

those. is equal to the product of the moved charge and the potential difference at the initial and final points.

The potential difference between two points of the electrostatic field φ 1 -φ 2 is equal to the voltage. Then

The ratio of the work done by the electrostatic field when moving a test charge from one point of the field to another to the value of this charge is calledvoltage between these points.

(12.30)

Graphically, the electric field can be represented not only using tension lines, but also using equipotential surfaces.

Equipotential surfaces – a set of points having the same potential. The figure shows that the tension lines (radial rays) are perpendicular to the equipotential lines.

E An infinite number of quipotential surfaces can be drawn around each charge and each system of charges (Fig. 12.14). However, they are carried out so that the potential differences between any two adjacent equipotential surfaces are the same. Then the density of equipotential surfaces clearly characterizes the field strength at different points. Where these surfaces are denser, the field strength is greater. Knowing the location of equipotential lines (surfaces), it is possible to construct tension lines, or based on the known location of tension lines, it is possible to construct equipotential surfaces.

§ 12.4The relationship between tension and potential

The electrostatic field has two characteristics: force (tension) and energy (potential). Tension and potential are different characteristics of the same point in the field, therefore, there must be a connection between them.

The work of moving a single point positive charge from one point to another along the x axis, provided that the points are located infinitely close to each other and x 1 – x 2 = dx, is equal to qE x dx. The same work is equal to q(φ 1 - φ 2)= -dφq. Equating both expressions, we can write

Repeating similar reasoning for the y and z axes, we can find the vector :

Where
- unit vectors of coordinate axes x, y, z.

From the definition of gradient it follows that

or
(12.31)

those. field strength E is equal to the potential gradient with a minus sign. The minus sign is determined by the fact that tension vector E field is directed towards decreasing potential.

The established connection between tension and potential allows us to find the potential difference between two arbitrary points of this field using a known field strength.

Field of a uniformly charged sphere radiusR

The field strength outside the sphere is determined by the formula

(r>R)

The potential difference between points r 1 and r 2 (r 1 >R; r 2 >R) is determined using the relation

We obtain the sphere potential if r 1 = R, r 2 → ∞:

Field of a uniformly charged infinitely long cylinder

The field strength outside the cylinder (r >R) is determined by the formula

(τ – linear density).

The potential difference between two points lying at a distance r 1 and r 2 (r 1 >R; r 2 >R) from the cylinder axis is equal to

(12.32)

Field of a uniformly charged infinite plane

The field strength of this plane is determined by the formula

(σ - surface density).

The potential difference between points lying at a distance x 1 and x 2 from the plane is equal to

(12.33)

Field of two oppositely charged infinite parallel planes

The field strength of these planes is determined by the formula

The potential difference between the planes is

(12.34)

(d – distance between planes).

Examples of problem solving

Example 12.1 . Three point charges Q 1 =2nC, Q 2 =3nC and Q 3 =-4nC are located at the vertices of an equilateral triangle with a side length a=10cm. Determine the potential energy of this system.

Given : Q 1 =2nC=2∙10 -9 C; Q 2 =3nC=3∙10 -9 C; and Q 3 =-4nC=4∙10 -9 C; a=10cm=0.1m.

Find : U.

R solution: The potential energy of a system of charges is equal to the algebraic sum of the interaction energies of each of the interacting pairs of charges, i.e.

U=U 12 +U 13 +U 23

where, respectively, the potential energies of one of the charges located in the field of another charge at a distance A from him are equal

;
;
(2)

Let us substitute formulas (2) into expression (1), and find the desired potential energy of the system of charges

Answer: U=-0.126 μJ.

Example 12.2 . Determine the potential in the center of a ring with an internal radius R 1 = 30 cm and an external radius R 2 = 60 cm, if a charge q = 5 nC is uniformly distributed on it.

Given: R 1 =30cm=0.3m; R 2 =60cm=0.6m; q=5nC=5∙10 -9 C

Find : φ .

Solution: Let us divide the ring into concentric infinitely thin rings with inner radius r and outer radius (r+dr).

The area of ​​the thin ring under consideration (see figure) dS=2πrdr.

P potential at the center of the ring, created by an infinitely thin ring,

where is the surface charge density.

To determine the potential at the center of the ring, one should arithmetically add dφ from all infinitely thin rings. Then

Considering that the ring charge Q=σS, where S= π(R 2 2 -R 1 2) is the area of ​​the ring, we obtain the desired potential in the center of the ring

Answer : φ=25V

Example 12.3. Two point charges of the same name (q 1 =2nC andq 2 =5nC) are in vacuum at a distancer 1 = 20cm. Determine the work A that must be done to bring them closer to the distancer 2 =5cm.

Given: q 1 =2nCl=2∙10 -9 Cl; q 2 =5nCl=5∙10 -9 Cl ; r 1 = 20cm=0.2m;r 2 =5cm=0.05m.

Find : A.

Solution: The work done by the forces of an electrostatic field when a charge Q moves from a field point with potential φ 1 to a point with potential φ 2.

A 12 = q(φ 1 - φ 2)

When charges of the same name come together, work is done by external forces, therefore the work of these forces is equal in magnitude, but opposite in sign to the work of Coulomb forces:

A= -q(φ 1 - φ 2)= q(φ 2 - φ 1). (1)

Potentials of points 1 and 2 of the electrostatic field

;
(2)

Substituting formulas (2) into expression (1), we find the required work that must be done to bring the charges closer together,

Answer: A=1.35 µJ.

Example 12.4. An electrostatic field is created by a positively charged endless thread. A proton moving under the influence of an electrostatic field along the tension line from a thread from a distancer 1 =2cm tor 2 =10cm, changed its speed fromυ 1 =1mm/s toυ 2 =5mm/s. Determine the linear charge density τ of the thread..

Given: q=1.6∙10 -19 C; m=1.67∙10 -27 kg; r 1 =2cm=2∙10 -2 m; r 2 = 10cm=0.1m; r 2 =5cm=0.05m; υ 1 =1Mm/s=1∙10 6 m/s; up to υ 2 =5Mm/s=5∙10 6 m/s.

Find : τ .

Solution: The work done by the forces of the electrostatic field when moving a proton from a field point with potential φ 1 to a point with potential φ 2 goes to increase the kinetic energy of the proton

q(φ 1 - φ 2)=ΔT (1)

In the case of a thread, the electrostatic field has axial symmetry, therefore

or dφ=-Edr,

then the potential difference between two points located at a distance r 1 and r 2 from the thread,

(take into account that the field strength created by a uniformly charged endless thread,
).

Substituting expression (2) into formula (1) and taking into account that
, we get

Where does the desired linear charge density of the thread come from?

Answer : τ = 4.33 µC/m.

Example 12.5. An electrostatic field is created in a vacuum by a ball of radiusR=8cm, uniformly charged with volume density ρ=10nC/m 3 . Determine the potential difference between two points of this field lying from the center of the ball at the distances: 1)r 1 =10cm andr 2 =15cm; 2)r 3 = 2cm andr 4 =5cm..

Given: R=8cm=8∙10 -2 m; ρ=10nC/m 3 =10∙10 -9 nC/m3; r 1 =10cm=10∙10 -2 m;

r 2 =15cm=15∙10 -2 m; r 3 = 2cm=2∙10 -2 m; r 4 =5cm=5∙10 -2 m.

Find : 1) φ 1 - φ 2 ; 2) φ 3 - φ 4 .

Solution: 1) The potential difference between two points located at a distance r 1 and r 2 from the center of the ball.

(1)

Where
is the field strength created by a uniformly charged ball with volume density ρ at any point lying outside the ball at a distance r from its center.

Substituting this expression into formula (1) and integrating, we obtain the desired potential difference

2) The potential difference between two points lying at a distance r 3 and r 4 from the center of the ball,

(2)

Where
is the field strength created by a uniformly charged ball with volume density ρ at any point lying inside the ball at a distance r from its center.

Substituting this expression into formula (2) and integrating, we obtain the desired potential difference

Answer : 1) φ 1 - φ 2 =0.643 V; 2) φ 3 - φ 4 =0.395 V

F is the force of interaction between two point charges

q 1 , q 2- magnitude of charges

ε α - absolute dielectric constant of the medium

r - distance between point charges

Conservative electrostatic interaction.

Let's calculate the work done by the electrostatic field created by the charge q´ by charge movement q from point 1 to point 2.

Work on the way d l is equal to:

where d r – radius vector increment when moving by d l; i.e.

Then the total work when moving q´ from point 1 to point 2 is equal to the integral:

The work of electrostatic forces does not depend on the shape of the path, but only on the coordinates of the starting and ending points of movement . Hence, field strengths are conservative, and the field itself – potentially.

Electrostatic field potential.

Electrostatic field potential - a scalar quantity equal to the ratio of the potential energy of a charge in the field to this charge:

Energy characteristics of the field at a given point. The potential does not depend on the amount of charge placed in this field.

Electrostatic field potential of a point charge.

Let us consider the special case when an electrostatic field is created by an electric charge Q. To study the potential of such a field, there is no need to introduce a charge q into it. You can calculate the potential of any point in such a field located at a distance r from the charge Q.

The dielectric constant of the medium has a known value (tabular) and characterizes the medium in which the field exists. For air it is equal to unity.

Formula for the operation of an electrostatic field.

A force acts on the charge q₀ from the field, which can do work and move this charge in the field.

The work of the electrostatic field does not depend on the trajectory. The work done by the field when a charge moves along a closed path is zero. For this reason, the forces of the electrostatic field are called conservative, and the field itself is called potential.

Relationship between electrostatic field strength and potential.

The intensity at any point of the electric field is equal to the potential gradient at this point, taken with the opposite sign. The minus sign indicates that the voltage E is directed in the direction of decreasing potential.

Electrical capacity of conductor and capacitor.

Electrical capacity - characteristic of a conductor, a measure of its ability to accumulate electrical charge

Formula for the electrical capacity of a flat capacitor.

Electric field energy.

Energy of a charged capacitor equal to the work of external forces that must be expended to charge the capacitor.

Electricity.

Electricity - directed (ordered) movement of charged particles

Conditions for the occurrence and existence of electric current.

1. presence of free charge carriers,

2. presence of potential difference. these are the conditions for the occurrence of current,

3. closed circuit,

4. a source of external forces that maintains a potential difference.

Outside forces.

Outside forces- forces of a non-electrical nature that cause the movement of electrical charges inside a direct current source. All forces other than Coulomb forces are considered external.

E.m.f. Voltage.

Electromotive force (EMF) - a physical quantity characterizing the work of third-party (non-potential) forces in direct or alternating current sources. In a closed conducting circuit, the EMF is equal to the work of these forces to move a single positive charge along the circuit.

EMF can be expressed in terms of the electric field strength of external forces

Voltage (U) equal to the ratio of the work of the electric field to move the charge
to the amount of charge moved in a section of the circuit.

SI unit of voltage:

Current strength.

Current strength (I)- a scalar quantity equal to the ratio of the charge q passing through the cross section of the conductor to the time period t during which the current flowed. The current strength shows how much charge passes through the cross section of the conductor per unit time.

Current density.

Current density j - a vector whose modulus is equal to the ratio of the current flowing through a certain area, perpendicular to the direction of the current, to the magnitude of this area.

The SI unit of current density is ampere per square meter (A/m2).