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 that moves the charge is variable and equal to

Where r is the variable distance between charges.

. This expression can be obtained like this:

The value is the potential energy W p of the charge at a given point of the electric field:

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

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

Then . For q = +1 .

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

The potential of an electric field point is equal to the work of moving a unit positive charge from a given point to infinity: . Unit of measurement - Volt \u003d 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 initial and final 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 the 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 parts of it are exposed to a constant electric field of high voltage.

A constant electric field during the procedure of general exposure 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, as it were, one of the capacitor plates, while the second is an electrode suspended above the head, or installed above the impact site at a distance of 6-10 cm. Under the influence of high voltage under the tips of the needles fixed on 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 vasculature. After a short-term vasospasm, 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 slowing of the pulse.

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

Examples of problem solving

1. During the operation of the franklinization apparatus, 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 the treatment session (15 min). The ionization potential of air molecules is considered equal to 13.54 V; conventionally, air is considered a homogeneous gas.

is the ionization potential, A is the work of ionization, N is the number of electrons.

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

From here

Electric dipole in medicine

According to Einthoven's theory underlying electrocardiography, the heart is an electric dipole located in the center of an equilateral triangle (Einthoven's triangle), the vertices of which can be conditionally considered

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

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 you to determine the relationship between the projections of the dipole moment of the heart on the sides of the triangle as follows:

Knowing the voltages U AB , U BC , U AC , one 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 the lead.

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

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

Lecture #4

contact phenomena

1. Contact potential difference. Laws of Volta.

2. Thermoelectricity.

3. Thermocouple, its use in medicine.

4. Potential of rest. Action potential and its distribution.

1. Contact potential difference. Laws of Volta.

In close contact of dissimilar metals, 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 to the metal. This work is called the work function of the electron from the metal.

Let us bring into contact two different metals 1 and 2, having the 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 >A1). 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 resulting potential difference creates an electric field of strength E, which makes it difficult to further "pump" the electrons and will completely stop it when the work of moving the electron due to the contact potential difference becomes equal to the work function difference:

(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 predominant transfer of free electrons from the first metal to the second will begin. As a result, the first metal will be positively charged, the second - negatively. There will be a potential difference between the metals, which will stop the further transfer of electrons. The resulting potential difference is determined by the expression:

, (2)

where k is Boltzmann's constant.

In the general case of the contact of metals that differ both in the work function and in the concentration of free electrons, the c.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 the second law of Volta.

If we now directly connect the end conductors, then the potential difference existing between them is compensated by an equal potential difference arising in contact 1 and 4. Therefore, the K.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 by different T a > T b . Then, according to formula (3), the f.r.p. in a hot junction more than in a cold junction: . As a result, a potential difference arises between junctions a and b, called thermoelectromotive force, and current I will flow in a 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 a current due to the difference in temperature of the contacts 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 0°C. It's called cold junction. The other junction is called the hot or measurement junction.

The thermocouple has significant advantages over mercury thermometers: it is sensitive, inertialess, allows measuring the temperature of small objects, and allows remote measurements.

Measurement of the profile of the temperature field of the human body.

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

Skin temperature has its own well-defined topography. The lowest temperature (23-30º) is in the distal limbs, the tip of the nose, and the auricles. The highest temperature is in the armpit, in the 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. The 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 propagation of an action potential 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 fundamental differences.

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

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

Thus, the propagation of the action potential along the nerve fiber occurs in the form of an autowave. Excitable cells are the active medium.

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 Ohm and a galvanometer with a resistance of r 2 = 80 Ohm are used; I=26 µA at junction temperature difference ºС. What is the thermocouple constant?

The thermopower that occurs in a thermocouple is , 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 #5

Electromagnetism

1. The nature of magnetism.

2. Magnetic interaction of currents in 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.

The magnetic field is a force field, it is depicted by means of magnetic lines of force. Unlike the lines of force of the electric field, magnetic lines of force are always closed.

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

2 . Magnetic interaction of currents in vacuum. Ampère's law.

The magnetic interaction of currents was studied using movable wire circuits. Ampere found that the magnitude of the force of interaction of two small sections of conductors 1 and 2 with currents is proportional to the lengths of these sections, the currents 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 the impact 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 coefficient of proportionality k, we obtain the mathematical expression of Ampère's law:

(3)

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

A current element is a vector equal in magnitude to the product Idl of an infinitely small section of the length dl of the conductor and the current strength I in it and directed along this current. Then, passing in (3) from small to infinitely small 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 as

. (6)

3 . Magnetic field strength. Ampere formula. Biot-Savart-Laplace law.

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

At the point O of this field, located at a distance r from dl, we place I 0 dl 0. Then, according to Ampère's law (6), this element will be affected by the force

(7)

where is the angle between the direction of the current I in the section dl (creating a field) and the direction of the radius vector r, and is the angle between the direction of the 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 as 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 the point O.

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

where is the angle between the direction of the current I 0 and the magnetic field dH. Formula (9) is called the Ampere formula, 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 “rule of the left hand”.

Assuming in (9) =90º, we get:

Those. the magnetic field strength is directed tangentially to the field line of force, and in magnitude it is equal 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 are called diamagnetic, the second - paramagnetic substances. Among paramagnets, 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 diamagnets, the total magnetic moment of an atom (molecule) is zero, because. the magnetic moments cancel each other out. However, under the influence of an external magnetic field, a magnetic moment is induced in these atoms, which is directed opposite to the external field. As a result, the diamagnetic medium becomes magnetized and creates its own magnetic field, directed oppositely to the external one and weakening it.

The induced magnetic moments of diamagnetic atoms are conserved as long as there is an external magnetic field. When the external field is eliminated, the induced magnetic moments of the atoms disappear and the diamagnet is demagnetized.

In paramagnetic atoms, the orbital, spin, nuclear moments do not compensate each other. However, the atomic magnetic moments are arranged randomly, so the paramagnetic medium does not exhibit magnetic properties. The external field rotates the atoms of the paramagnet so that their magnetic moments are set predominantly in the direction of the field. As a result, the paramagnet is magnetized and creates its own magnetic field, coinciding with the external one and amplifying 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, a ferromagnet is not magnetized.

With the advent 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; ferromagnet is magnetized. At a sufficiently strong field, all domains are reoriented along the field, and the ferromagnet is rapidly magnetized to saturation.

When the external field is eliminated, the ferromagnet is not completely demagnetized, but retains the residual magnetic induction, since thermal motion cannot misorient 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 able to disorient the atoms in the 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)

The unit of measure B is Tesla, F-Weber.

Conduction electrons do not spontaneously leave the metal in a noticeable amount. This is explained by the fact that the metal represents a potential well for them. It is possible to leave the metal only for those electrons whose energy is sufficient to overcome the potential barrier existing on the surface. The forces that cause this barrier have the following origin. The accidental removal of an electron from the outer layer of positive ions of the lattice leads to the appearance of an excess positive charge in the place that the electron left.

The Coulomb interaction with this charge causes the electron, whose speed is not very high, to return back. Thus, individual electrons leave the metal surface all the time, move away from it by several interatomic distances, and then turn back. As a result, the metal is surrounded by a thin cloud of electrons. This cloud, together with the outer layer of ions, forms a double electric layer (Fig. 60.1; circles - ions, black dots - electrons). The forces acting on an electron in such a layer are directed inside the metal.

The work done against these forces when transferring an electron from the metal to the outside does not increase the potential energy of the electron

Thus, the potential energy of valence electrons inside the metal is less than outside the metal by an amount equal to the depth of the potential well (Fig. 60.2). The change in energy occurs over a length of the order of several interatomic distances; therefore, the walls of the well can be considered vertical.

The potential energy of an electron and the potential of the point where the electron is located have opposite signs. It follows that the potential inside the metal is greater than the potential in the immediate vicinity of its surface (we will simply say “on the surface” for brevity) by the value

Giving excess positive charge to the metal increases the potential both on the surface and inside the metal. The potential energy of the electron decreases accordingly (Fig. 60.3, a).

Recall that the values ​​of the potential and potential energy at infinity are taken as the origin. The introduction of a negative charge lowers the potential inside and outside the metal. Accordingly, the potential energy of the electron increases (Fig. 60.3, b).

The total energy of an electron in a metal is the sum of the potential and kinetic energies. In § 51 it was clarified that at absolute zero the values ​​of the kinetic energy of the conduction electrons range from zero to the energy Emax coinciding with the Fermi level. On fig. 60.4 the energy levels of the conduction band are inscribed in the potential well (dotted lines show unoccupied levels). To move out of the metal, different electrons need to be given different energies.

Thus, an electron located at the lowest level of the conduction band must be given energy for an electron located at the Fermi level, the energy is sufficient

The smallest energy that needs to be imparted to an electron in order to remove it from a solid or liquid body into a vacuum is called the work function. The work function is usually denoted by where Ф is a value called the output potential.

In accordance with the above, the work function of an electron from a metal is determined by the expression

We obtained this expression under the assumption that the temperature of the metal is 0 K. At other temperatures, the work function is also defined as the difference between the depth of the potential well and the Fermi level, i.e., definition (60.1) is extended to any temperature. The same definition applies to semiconductors.

The Fermi level depends on temperature (see formula (52.10)). In addition, due to the change in the average distances between atoms due to thermal expansion, the depth of the potential well changes slightly. This leads to the fact that the work function is slightly dependent on temperature.

The work function is very sensitive to the state of the metal surface, in particular to its purity. By choosing the right surface coating, the work function can be greatly reduced. So, for example, applying a layer of alkaline earth metal oxide (Ca, Sr, Ba) to the surface of tungsten reduces the work function from 4.5 eV (for pure W) to 1.5-2.

Metals contain conduction electrons that form an electron gas and participate in thermal motion. Since the conduction electrons are kept inside the metal, then, therefore, near the surface there are forces acting on the electrons and directed inside the metal. In order for an electron to leave the metal beyond its limits, a certain work A must be done against these forces, which is called the work function of the electron leaving the metal. This work, of course, is different for different metals.

The potential energy of an electron inside a metal is constant and equal to:

Wp = -eφ , where j is the potential of the electric field inside the metal.

21. Contact potential difference - this is the potential difference between conductors that occurs when two different conductors that have the same temperature come into contact.

When two conductors with different work functions come into contact, electric charges appear on the conductors. And between their free ends there is a potential difference. The potential difference between points located outside the conductors, near their surface is called the contact potential difference. Since the conductors are at the same temperature, in the absence of an applied voltage, the field can only exist in the boundary layers (Volta's Rule). Distinguish between internal potential difference (when metals come into contact) and external (in the gap). The value of the external contact potential difference is equal to the difference in the work functions per electron charge. If the conductors are connected into a ring, then the EMF in the ring will be 0. For different pairs of metals, the value of the contact potential difference ranges from tenths of a volt to a few volts.

The operation of a thermoelectric generator is based on the use of the thermoelectric effect, the essence of which lies in the fact that when the junction (junction) of two different metals is heated between their free ends, which have a lower temperature, a potential difference arises, or the so-called thermoelectromotive force (thermo-EMF). If such a thermoelement (thermocouple) is connected to external resistance, then an electric current will flow through the circuit (Fig. 1). Thus, in thermoelectric phenomena, there is a direct conversion of thermal energy into electrical energy.

The value of the thermoelectromotive force is determined approximately by the formula E \u003d a (T1 - T2)

22. A magnetic field - a force field acting on moving electric charges and on bodies with a magnetic moment, regardless of the state of their movement; magnetic component of the electromagnetic field

moving charge q, creates a magnetic field around itself, the induction of which

where is the electron velocity, is the distance from the electron to a given point of the field, μ is the relative magnetic permeability of the medium, μ 0 = 4π 10 -7 H/m is the magnetic constant.

Magnetic induction- vector quantity, which is a force characteristic of the magnetic field (its action on charged particles) at a given point in space. Determines the force with which the magnetic field acts on a charge moving at speed.

More specifically, is such a vector that the Lorentz force acting from the side of the magnetic field on a charge moving at speed is equal to

23. According to the Biot-Savart-Laplace law contour element dl through which current flows I, creates a magnetic field around itself, the induction of which at some point K

where is the distance from the point K to current element dl, α is the angle between the radius vector and the current element dl.

The direction of the vector can be found from Maxwell's rule(gimlet): if you screw in a gimlet with a right-hand thread in the direction of current in the conductor element, then the direction of movement of the gimlet handle will indicate the direction of the magnetic induction vector.

Applying the Biot-Savart-Laplace law to contours of various types, we obtain:

in the center of a circular loop of radius R with current power I magnetic induction

magnetic induction on the axis of circular current Where a is the distance from the point where you are looking for B to the plane of the circular current,

the field created by an infinitely long current-carrying conductor at a distance r from conductor

the field created by a conductor of finite length, at a distance r from the conductor (Fig. 15)

field inside a toroid or an infinitely long solenoid n- the number of turns per unit length of the solenoid (toroid)

The magnetic induction vector is related to the magnetic field strength by the relation

Bulk energy density magnetic field:

25 .On a charged particle moving in a magnetic field with induction B with speed υ , from the side of the magnetic field there is a force called Lorentz force

and the modulus of this force is equal to .

The direction of the Lorentz force can be determined from left hand rule: if you put your left hand so that the perpendicular to the velocity component of the induction vector enters the palm, and four fingers are located in the direction of the velocity of the positive charge (or against the direction of the velocity of the negative charge), then the bent thumb will indicate the direction of the Lorentz force

26 .The principle of operation of cyclic charged particle accelerators.

The independence of the rotation period T of a charged particle in a magnetic field was used by the American scientist Lawrence in the idea of ​​a cyclotron - an accelerator of charged particles.

Cyclotron consists of two dees D 1 and D 2 - hollow metal half-cylinders placed in a high vacuum. An accelerating electric field is created in the gap between the dees. A charged particle entering this gap increases its speed and flies into the space of a half-cylinder (dee). The dees are placed in a constant magnetic field, and the trajectory of the particle inside the dee will be curved in a circle. When the particle enters the gap between the dees for the second time, the polarity of the electric field changes and it again becomes accelerating. An increase in speed is accompanied by an increase in the radius of the trajectory. In practice, an alternating field is applied to the dees with a frequency ν= 1/T=(B/2π)(q/m) . The speed of the particle increases each time in the gap between the dees under the action of an electric field.

27.Amp power is the force acting on a conductor carrying current I located in a magnetic field

Δ l- the length of the conductor, and the direction coincides with the direction of current in the conductor.

Ampere Power Module: .

Two parallel infinitely long straight conductors with currents I 1 And I 2 interact with each other with force

Where l- the length of the conductor section, r- distance between conductors.

28. Interaction of parallel currents - Ampère's law

Now you can easily get a formula for calculating the interaction force of two parallel currents.

So, along two long straight parallel conductors (Fig. 440), located at a distance R from each other (which is much, 15 times less than the length of the conductors), direct currents I 1, I 2 flow.

In accordance with field theory, the interaction of conductors is explained as follows: the electric current in the first conductor creates a magnetic field that interacts with the electric current in the second conductor. To explain the emergence of a force acting on the first conductor, it is necessary to “reverse roles” of the conductors: the second one creates a field that acts on the first one. Mentally turn the right screw, turn it with your left hand (or use the vector product) and make sure that with currents flowing in one direction, the conductors attract, and with currents flowing in opposite directions, the conductors repel1.

Thus, the force acting on a section of length Δl of the second conductor is the Ampère force, it is equal to

where B1 are the inductions of the magnetic field created by the first conductor. When writing this formula, it is taken into account that the induction vector B1 is perpendicular to the second conductor. The induction of the field created by direct current in the first conductor, at the location of the second, is equal to

From formulas (1), (2) it follows that the force acting on the selected section of the second conductor is equal to

29. A coil with current in a magnetic field.

If we place in a magnetic field not a conductor, but a coil (or coil) with a current and place it vertically, then, applying the left hand rule to the upper and lower sides of the coil, we get that the electromagnetic forces F acting on them will be directed in different directions. As a result of the action of these two forces, an electromagnetic torque M is generated, which will cause the coil to rotate, in this case clockwise. This moment

where D is the distance between the sides of the coil.

The coil will rotate in the magnetic field until it takes a position perpendicular to the magnetic field lines (Fig. 50, b). In this position, the greatest magnetic flux will pass through the coil. Consequently, a coil or coil with current introduced into an external magnetic field always tends to take such a position that the largest possible magnetic flux passes through the coil.

Magnetic moment, magnetic dipole moment- the main quantity that characterizes the magnetic properties of a substance (according to the classical theory of electromagnetic phenomena, the source of magnetism is electrical macro- and microcurrents; a closed current is considered an elementary source of magnetism). Elementary particles, atomic nuclei, electron shells of atoms and molecules have a magnetic moment. The magnetic moment of elementary particles (electrons, protons, neutrons and others), as shown by quantum mechanics, is due to the existence of their own mechanical moment - spin.

30. magnetic flux - a physical quantity equal to the flux density of field lines passing through an infinitely small area dS. Flow F in as an integral of the magnetic induction vector IN through a finite surface S Defined in terms of an integral over a surface.

31. Work on moving a conductor with current in a magnetic field

Consider a current-carrying circuit formed by fixed wires and a movable jumper of length l sliding along them (Fig. 2.17). This contour is located in an external uniform magnetic field perpendicular to the plane of the contour.

The current element I (movable wire) of length l is affected by the Ampere force directed to the right:

Let the conductor l move parallel to itself at a distance dx. This will do the work:

dA=Fdx=IBldx=IBdS=IdФ

The work done by a conductor with current when moving is numerically equal to the product of the current and the magnetic flux crossed by this conductor.

The formula remains valid if a conductor of any shape moves at any angle to the lines of the magnetic induction vector.

32. Matter magnetization . Permanent magnets can be made from only relatively few substances, but all substances placed in a magnetic field become magnetized, that is, they themselves become sources of a magnetic field. As a result, the vector of magnetic induction in the presence of matter differs from the vector of magnetic induction in vacuum.

The magnetic moment of an atom is composed of the orbital and intrinsic moments of the electrons included in its composition, as well as the magnetic moment of the nucleus (which is due to the magnetic moments of the elementary particles that make up the nucleus - protons and neutrons). The magnetic moment of the nucleus is much less than the moments of the electrons; therefore, when considering many issues, it can be neglected and it can be assumed that the magnetic moment of an atom is equal to the vector sum of the magnetic moments of electrons. The magnetic moment of a molecule can also be considered equal to the sum of the magnetic moments of its constituent electrons.

Thus, an atom is a complex magnetic system, and the magnetic moment of the atom as a whole is equal to the vector sum of the magnetic moments of all electrons

Magnetics and they call substances that can be magnetized in an external magnetic field, i.e. capable of creating their own magnetic field. The intrinsic field of substances depends on the magnetic properties of their atoms. In this sense, magnets are magnetic analogues of dielectrics.

According to classical concepts, an atom consists of electrons moving in orbits around a positively charged nucleus, which, in turn, consists of protons and neutrons.

All substances are magnets, i.e. All substances are magnetized in an external magnetic field, but their nature and degree of magnetization are different. Depending on this, all magnets are divided into three types: 1) diamagnets; 2) paramagnets; 3) ferromagnets.

Diamagnets. - it includes many metals (for example, copper, zinc, silver, mercury, bismuth), most gases, phosphorus, sulfur, quartz, water, the vast majority of organic compounds, etc.

Diamagnets have the following properties:

2) own magnetic field is directed against the external one and slightly weakens it (m<1);

3) there is no residual magnetism (the intrinsic magnetic field of a diamagnet disappears after the external field is removed).

The first two properties indicate that the relative magnetic permeability m of diamagnets is only slightly less than 1. For example, the strongest of diamagnets, bismuth, has m = 0.999824.

Paramagnets- they include alkali and alkaline earth metals, aluminum, tungsten, platinum, oxygen, etc.

Paramagnets have the following properties:

1) very weak magnetization in an external magnetic field;

2) own magnetic field is directed along the external one and amplifies it a little (m>1);

3) no residual magnetism.

It follows from the first two properties that the value of m is only slightly greater than 1. For example, for one of the strongest paramagnetic materials, platinum, the relative magnetic permeability m=1.00036.

33.ferromagnets - these include iron, nickel, cobalt, gadolinium, their alloys and compounds, as well as some alloys and compounds of manganese and chromium with non-ferromagnetic elements. All these substances have ferromagnetic properties only in the crystalline state.

Ferromagnets have the following properties:

1) very strong magnetization;

2) the own magnetic field is directed along the external one and significantly enhances it (the values ​​of m range from several hundred to several hundred thousand);

3) relative magnetic permeability m depends on the magnitude of the magnetizing field;

4) there is residual magnetism.

Domain- a macroscopic region in a magnetic crystal, in which the orientation of the vector of spontaneous homogeneous magnetization or the vector of antiferromagnetism (at a temperature below the Curie or Neel point, respectively) in a certain - strictly ordered - way is rotated or shifted, that is, polarized, relative to the directions of the corresponding vector in neighboring domains.

Domains are formations consisting of a huge number of [ordered] atoms and sometimes visible to the naked eye (dimensions of the order of 10−2 cm3).

Domains exist in ferro- and antiferromagnetic, ferroelectric crystals and other substances with spontaneous long-range order.

Curie point or Curie temperature- the temperature of the phase transition of the second kind, associated with an abrupt change in the properties of the symmetry of a substance (for example, magnetic - in ferromagnets, electric - in ferroelectrics, crystal chemical - in ordered alloys). Named after P. Curie. At a temperature T below the Curie point Q, ferromagnets have spontaneous (spontaneous) magnetization and a certain magneto-crystalline symmetry. At the Curie point (T=Q), the intensity of the thermal motion of ferromagnet atoms is sufficient to destroy its spontaneous magnetization (“magnetic order”) and change the symmetry, as a result, the ferromagnet becomes a paramagnet. Similarly, for antiferromagnets at T=Q (at the so-called antiferromagnetic Curie point or Neel point), the destruction of their characteristic magnetic structure (magnetic sublattices) occurs, and antiferromagnets become paramagnets. In ferroelectrics and antiferroelectrics at T=Q, the thermal motion of atoms nullifies the spontaneous ordered orientation of the electric dipoles of the elementary cells of the crystal lattice. In ordered alloys, at the Curie point (it is also called a point in the case of alloys.

Magnetic hysteresis observed in magnetically ordered substances (in a certain temperature range), for example, in ferromagnets, usually divided into domains of the region of spontaneous (spontaneous) magnetization, in which the magnetization value (magnetic moment per unit volume) is the same, but the directions are different.

Under the action of an external magnetic field, the number and size of domains magnetized in the field increase at the expense of other domains. The magnetization vectors of individual domains can rotate along the field. In a sufficiently strong magnetic field, a ferromagnet is magnetized to saturation, while it consists of a single domain with saturation magnetization JS directed along the external field H.

Typical dependence of the magnetization on the magnetic field in the case of hysteresis

34. Earth's magnetic field

As you know, a magnetic field is a special kind of force field that affects bodies with magnetic properties, as well as moving electric charges. To a certain extent, the magnetic field can be considered a special kind of matter that transmits information between electric charges and bodies with a magnetic moment. Accordingly, the Earth's magnetic field is a magnetic field that is created due to factors associated with the functional features of our planet. That is, the geomagnetic field is created by the Earth itself, and not by external sources, although the latter have a certain effect on the planet's magnetic field.

Thus, the properties of the Earth's magnetic field inevitably depend on the features of its origin. The main theory explaining the emergence of this force field is associated with the flow of currents in the liquid metal core of the planet (the temperature at the core is so high that the metals are in a liquid state). The energy of the Earth's magnetic field is generated by the so-called hydromagnetic dynamo mechanism, which is due to the multidirectionality and asymmetry of electric currents. They generate amplification of electrical discharges, which leads to the release of thermal energy and the emergence of new magnetic fields. It is curious that the mechanism of the hydromagnetic dynamo has the ability to “self-excite”, that is, the active electrical activity inside the earth's core constantly generates a geomagnetic field without external influence.

35.Magnetization - vector physical quantity characterizing the magnetic state of a macroscopic physical body. It is usually denoted M. It is defined as the magnetic moment of a unit volume of a substance:

Here, M is the magnetization vector; - vector of the magnetic moment; V - volume.

In the general case (the case of an inhomogeneous, for one reason or another, medium), the magnetization is expressed as

and is a function of the coordinates. Where is the total magnetic moment of the molecules in the volume dV The relationship between M and the magnetic field strength H in diamagnetic and paramagnetic materials is usually linear (at least for not too large values ​​of the magnetizing field):

where χm is called the magnetic susceptibility. In ferromagnetic materials, there is no one-to-one relationship between M and H due to magnetic hysteresis, and the magnetic susceptibility tensor is used to describe the dependence.

Magnetic field strength(standard designation H) - a vector physical quantity equal to the difference between the magnetic induction vector B and the magnetization vector M.

In the International System of Units (SI): H = (1/µ 0)B - M where µ 0 is the magnetic constant.

Magnetic permeability- physical quantity, coefficient (depending on the properties of the medium), characterizing the relationship between the magnetic induction B and the magnetic field strength H in the substance. For different media, this coefficient is different, so they talk about the magnetic permeability of a particular medium (implying its composition, state, temperature, etc.).

Usually denoted by the Greek letter µ. It can be either a scalar (for isotropic substances) or a tensor (for anisotropic substances).

In general, the relationship relationship between magnetic induction and magnetic field strength through magnetic permeability is introduced as

and in the general case it should be understood here as a tensor, which in the component notation corresponds to

Control questions .. 18

9. Laboratory work No. 2. Study of thermionic emission at low emission current densities . 18

Work order .. 19

Report Requirements . 19

Control questions .. 19

Introduction

Emission electronics studies the phenomena associated with the emission (emission) of electrons from a condensed medium. Electronic emission occurs when a part of the body's electrons acquires, as a result of external influence, energy sufficient to overcome the potential barrier at its boundary, or if an external electric field makes it "transparent" for a part of the electrons. Depending on the nature of the external influence, there are:

• thermionic emission (heating of bodies);
• secondary electron emission (electron bombardment of the surface);
• ion-electron emission (surface bombardment with ions);
• photoelectronic emission (electromagnetic radiation);
• exoelectronic emission (mechanical, thermal and other types of surface treatment);
• field emission (external electric field), etc.

In all phenomena where it is necessary to take into account either the exit of an electron from a crystal into the surrounding space, or the transition from one crystal to another, the characteristic called the "Work function" becomes decisive. The work function is defined as the minimum energy required to extract an electron from a solid and place it at a point where its potential energy is conventionally taken to be zero. In addition to describing various emission phenomena, the concept of work function plays an important role in explaining the occurrence of a contact potential difference in the contact of two metals, a metal with a semiconductor, two semiconductors, as well as galvanic phenomena.

The guidelines consist of two parts. The first part contains basic theoretical information on emission phenomena in solids. The main attention is paid to the phenomenon of thermionic emission. The second part contains a description of laboratory works devoted to the experimental study of thermionic emission, the study of the contact potential difference and the distribution of the work function over the surface of the sample.

Part 1. Basic theoretical information

1. Work function of an electron. Effect on work output of surface condition

The fact that the electrons are kept inside the solid indicates that a retarding field arises in the surface layer of the solid, which prevents the electrons from leaving it into the surrounding vacuum. A schematic representation of a potential barrier at the boundary of a solid body is given in fig. 1. To leave the crystal, an electron must do work equal to the work function. Distinguish thermodynamic And external exit work.

The thermodynamic work function is the difference between the energy of the zero vacuum level and the Fermi energy of a solid.

The external work function (or electron affinity) is the difference between the energy of the zero vacuum level and the energy of the bottom of the conduction band (Fig. 1).

Rice. 1. Shape of the crystal potential U along the line of location of ions in the crystal and in the near-surface region of the crystal: the positions of the ions are marked with dots on the horizontal line; φ=- U /e is the work function potential; E F is the Fermi energy (negative); E Cis the energy of the bottom of the conduction band; WO is the thermodynamic work function; Wa is the external work function; the shaded area conventionally depicts filled electronic states

Two main reasons for the appearance of a potential barrier at the interface between a solid body and a vacuum can be indicated. One of them is related to the fact that an electron emitted from a crystal induces a positive electric charge on its surface. An attractive force arises between the electron and the surface of the crystal (the force of the electric image, see Section 5, Fig. 12), which tends to return the electron back to the crystal. Another reason is related to the fact that electrons due to thermal motion can cross the surface of the metal and move away from it for small distances (on the order of atomic). They form a negatively charged layer above the surface. In this case, after the release of electrons, a positively charged layer of ions is formed on the crystal surface. As a result, an electrical double layer is formed. It does not create a field in the outer space, but it also requires work to overcome the electric field inside the double layer itself.

The value of the work function for most metals and semiconductors is a few electron volts. For example, for lithium, the work function is 2.38 eV, iron - 4.31 eV, germanium - 4.76 eV, silicon - 4.8 eV. To a large extent, the value of the work function is determined by the crystallographic orientation of the single crystal face from which electron emission occurs. For the (110) plane of tungsten, the work function is 5.3 eV; for the (111) and (100) planes, these values ​​are 4.4 eV and 4.6 eV, respectively.

The work function is greatly affected by thin layers deposited on the crystal surface. Atoms or molecules that settle on the surface of a crystal often donate an electron to it or take an electron from it and become ions. On fig. 2 shows the energy diagram of a metal and an isolated atom for the case when the thermodynamic work function of an electron from the metal W0 more than the ionization energy E ion deposited on its surface of the atom, In this situation, the electron of the atom is energetically favorable tunnel into the metal and descend in it to the Fermi level. The surface of a metal covered with such atoms is negatively charged and forms a double electric layer with positive ions, the field of which will reduce the work function of the metal. On fig. 3a shows a tungsten crystal coated with a cesium monolayer. Here the situation discussed above is realized, since the energy E ion cesium (3.9 eV) is less than the work function of tungsten (4.5 eV). In experiments, the work function decreases by more than three times. The opposite situation is observed if tungsten is covered with oxygen atoms (Fig. 3b). Since the bond of valence electrons in oxygen is stronger than in tungsten, when oxygen is adsorbed on the surface of tungsten, a double electric layer is formed, which increases the work function of the metal. The most common case is when an atom that has settled on the surface does not completely give up its electron to the metal or accepts an extra electron, but deforms its electron shell so that the atoms adsorbed on the surface become polarized and become electric dipoles (Fig. 3c). Depending on the orientation of the dipoles, the work function of the metal decreases (the orientation of the dipoles corresponds to Fig. 3c) or increases.

2. The phenomenon of thermionic emission

Thermionic emission is one of the types of electron emission by a solid surface. In the case of thermionic emission, the external action is associated with the heating of the solid.

The phenomenon of thermionic emission is the emission of electrons by heated bodies (emitters) into a vacuum or other medium.

Under conditions of thermodynamic equilibrium, the number of electrons n(E) having energy in the range from E before E+d E, is determined by the Fermi-Dirac statistics:

,(1)

Where g(E) is the number of quantum states corresponding to the energy E; E F is the Fermi energy; k is the Boltzmann constant; T is the absolute temperature.

On fig. 4 shows the energy scheme of the metal and the energy distribution curves of electrons at T\u003d 0 K, at low temperature T 1 and at high temperature T 2. At 0 K the energy of all electrons is less than the Fermi energy. None of the electrons can leave the crystal and no thermionic emission is observed. With an increase in temperature, the number of thermally excited electrons that can leave the metal increases, which causes the phenomenon of thermionic emission. On fig. 4 this is illustrated by the fact that T=T 2 The "tail" of the distribution curve goes beyond the zero level of the potential well. This indicates the appearance of electrons with energies exceeding the height of the potential barrier.

For metals, the work function is a few electron volts. Energy k T even at a temperature of thousands of Kelvins, it is a fraction of an electron volt. For pure metals, a significant emission of electrons can be obtained at a temperature of the order of 2000 K. For example, in pure tungsten, a noticeable emission can be obtained at a temperature of 2500 K.

To study thermionic emission, it is necessary to create an electric field near the surface of a heated body (cathode), which accelerates electrons for their removal (suction) from the emitter surface. Under the action of an electric field, the emitted electrons begin to move and an electric current is formed, which is called thermionic. To observe the thermionic current, a vacuum diode is usually used - an electron lamp with two electrodes. The cathode of the lamp is a filament made of refractory metal (tungsten, molybdenum, etc.), heated by an electric current. The anode is usually in the form of a metal cylinder surrounding an incandescent cathode. To observe the thermionic current, the diode is connected to the circuit shown in Fig. 5. It is obvious that the strength of the thermionic current should increase with an increase in the potential difference V between anode and cathode. However, this increase is not proportional V(Fig. 6). Upon reaching a certain voltage, the growth of the thermionic current practically stops. The limiting value of the thermionic current at a given cathode temperature is called the saturation current. The value of the saturation current is determined by the number of thermoelectrons that are able to leave the cathode surface per unit time. In this case, all the electrons supplied as a result of thermionic emission from the cathode are used to generate an electric current.

3. Dependence of thermionic current on temperature. Formula Richardson-Deshman

When calculating the thermionic current density we will use the electron gas model and apply to it the Fermi-Dirac statistics. Obviously, the density of the thermionic current is determined by the density of the electron cloud near the crystal surface, which is described by formula (1). Let us pass in this formula from the electron energy distribution to the momentum distribution of electrons. At the same time, we take into account that the allowed values ​​of the electron wave vector k V k -space are distributed uniformly so that for each value k accounts volume 8 p 3 (for a crystal volume equal to unity). Considering that the momentum of the electron p =ћ k we obtain that the number of quantum states in the volume element of the momentum space dp xdpydpz will be equal to

(2)

The two in the numerator of formula (2) takes into account two possible values ​​of the electron spin.

Let's direct the axis z rectangular coordinate system normal to the cathode surface (Fig. 7). Let us allocate an area of ​​unit area on the surface of the crystal and construct on it, as on the basis, a rectangular parallelepiped with a side edge vz =p z /m n(m n is the effective electron mass). Electrons contribute to the saturation current density by the component vz axis speed z. The contribution to the current density from one electron is

(3)

Where e is the charge of an electron.

The number of electrons in the parallelepiped whose velocities are contained in the interval under consideration:

In order for the crystal lattice not to be destroyed during the emission of electrons, an insignificant part of the electrons must come out of the crystal. For this, as formula (4) shows, the condition HERF>> k T. For such electrons, the unit in the denominator of formula (4) can be neglected. Then this formula is transformed to the form

(5)

Find now the number of electrons dN in the volume under consideration z- component of the momentum of which is enclosed between R z And R z +dpz. To do this, the previous expression must be integrated over R x And R y ranging from –∞ to +∞. When integrating it should be taken into account that

,

and use the table integral

,.

As a result, we get

.(6)

Now, taking into account (3), we find the density of the thermionic current created by all the electrons of the parallelepiped. To do this, expression (6) must be integrated for all electrons whose kinetic energy is at the Fermi level E≥E F+W0.Only such electrons can leave the crystal and only they play a role in the calculation of the thermal current. The component of the momentum of such electrons along the axis Z must satisfy the condition

Therefore, the saturation current density

Integration is performed for all values ​​of . We introduce a new integration variable

Then p z dp z =m n du And

.(8)

As a result, we get

,(9)

,(10)

where is the constant

.

Equality (10) is called the formula Richardson-Deshman. By measuring the density of the saturation thermionic current, one can use this formula to calculate the constant A and the work function W 0 . For experimental calculations, the formula Richardson-Deshman it is convenient to represent in the form

In this case, on the graph, the dependence ln(js /T2) from 1 /T expressed as a straight line. The intersection of the line with the y-axis calculates ln A , and the work function is determined from the slope of the straight line (Fig. 8).

4. Contact potential difference

Consider the processes that occur when two electronic conductors, for example, two metals, with different work functions approach and come into contact. Energy schemes of these metals are shown in fig. 9. Let EF1 And EF2 are the Fermi energy for the first and second metal, respectively, and W01 And W02 is their work function. In an isolated state, metals have the same vacuum level and, therefore, different Fermi levels. Assume for definiteness that W01< W02, then the Fermi level of the first metal will be higher than that of the second (Fig. 9a). When these metals come into contact against the occupied electronic states in metal 1, there are free energy levels of metal 2. Therefore, when these conductors come into contact, a resulting flow of electrons occurs from conductor 1 to conductor 2. This leads to the fact that the first conductor, losing electrons, becomes positively charged, and the second conductor, acquiring additional negative charge is negatively charged. Due to charging, all energy levels of metal 1 are shifted down, and metal 2 - up. The process of shifting the levels and the process of transition of electrons from conductor 1 to conductor 2 will continue until the Fermi levels of both conductors are aligned (Fig. 9b). As can be seen from this figure, the equilibrium state corresponds to the potential difference between the zero levels of the conductors 0 1 and 0 2:

.(11)

Potential difference V K.R.P called contact potential difference. Therefore, the contact potential difference is determined by the difference in the work functions of the electrons from the contacting conductors. The result obtained is valid for any methods of electron exchange between two materials, including thermionic emission in vacuum, through an external circuit, etc. Similar results are obtained when a metal contacts a semiconductor. A contact potential difference arises between the metals and the semiconductor, having approximately the same order of magnitude as in the case of contact between two metals (approximately 1 V). The only difference is that if in conductors the entire contact potential difference falls practically on the gap between the metals, then when the metal contacts the semiconductor, the entire contact potential difference falls on the semiconductor, in which a sufficiently large layer is formed, enriched or depleted in electrons. If this layer is depleted in electrons (in the case when the work function of the n-type semiconductor is less than the work function of the metal), then such a layer is called blocking and such a transition will have straightening properties. The potential barrier that occurs in the rectifying contact of a metal with a semiconductor is called Schottky barrier, and the diodes working on its basis - Schottky diodes.

Volt-amperecharacteristic of the hot cathode at low emission current densities. Schottky effect

If between the thermal cathode and the anode of the diode (Fig. 5) create a potential difference V, which prevents the movement of electrons to the anode, then only those electrons that flew out of the cathode with a kinetic energy reserve not less than the energy of the electrostatic field between the anode and cathode can get to the anode, i.e. –e V(V< 0). To do this, their energy in the hot cathode must be at least W 0 -eV. Then, replacing in the formula Richardson-Deshman (10) W0 on W 0 -eV, we obtain the following expression for the thermal emission current density:

,(12)

Here j S is the saturation current density. We logarithm this expression

.(13)

With a positive potential at the anode, all electrons leaving the hot cathode go to the anode. Therefore, the current in the circuit should not change, remaining equal to the saturation current. Thus, volt-ampere characteristic (CVC) of the thermal cathode will have the form shown in fig. 10 (curve a).

A similar I–V characteristic is observed only at relatively low emission current densities and high positive potentials at the anode, when there is no significant space charge of electrons near the emitting surface. The volt-ampere characteristic of the hot cathode, taking into account the space charge, considered in Sec. 6.

Let us note one more important feature of the CVC at low emission current densities. The conclusion that the thermal current reaches saturation at V=0 is valid only for the case when the cathode and anode materials have the same thermodynamic work function. If the work functions of the cathode and anode are not equal to each other, then a contact potential difference appears between the anode and cathode. In this case, even in the absence of an external electric field ( V= 0) between the anode and cathode there is an electric field due to the contact potential difference. For example, if W 0k< W 0a the anode will be negatively charged relative to the cathode. To destroy the contact potential difference, a positive bias should be applied to the anode. That's why volt-ampere the characteristic of the hot cathode is shifted by the value of the contact potential difference towards the positive potential (Fig. 10, curve b). With an inverse relationship between W 0k And W 0a the direction of the CVC shift is opposite (curve c in Fig. 10).

Conclusion about the independence of the saturation current density at V>0 is highly idealized. In real I–V characteristics of thermionic emission, a slight increase in thermionic emission current is observed with increasing V in saturation mode, which is associated with Schottky effect(Fig. 11).

The Schottky effect is a decrease in the work function of electrons from solids under the action of an external accelerating electric field.

To explain the Schottky effect, consider the forces acting on an electron near the crystal surface. In accordance with the law of electrostatic induction, surface charges of the opposite sign are induced on the surface of the crystal, which determine the interaction of the electron with the surface of the crystal. In accordance with the method of electrical images, the action of real surface charges on an electron is replaced by the action of a fictitious point positive charge +e located at the same distance from the crystal surface as the electron, but on the opposite side of the surface (Fig. 12). Then, in accordance with Coulomb's law, the force of interaction of two point charges

,(14)

Here ε o– electrical constant: X is the distance between the electron and the crystal surface.

The potential energy of an electron in the electric image force field, if counted from the zero vacuum level, is equal to

.(15)

Potential energy of an electron in an external accelerating electric field E

Total potential energy of an electron

.(17)

A graphical finding of the total energy of an electron located near the surface of a crystal is shown in fig. 13, which clearly shows the decrease in the work function of the electron from the crystal. The total curve of the potential energy of an electron (solid curve in Fig. 13) reaches a maximum at the point x m:

.(18)

This point is 10Å away from the surface at an external field strength » 3× 10 6 V/cm.

At the point X m the total potential energy equal to the potential barrier reduction (and hence the work function reduction),

.(19)

As a result of the Schottky effect, the current of the thermal diode at a positive voltage at the anode increases with an increase in the anode voltage. This effect manifests itself not only when electrons are emitted into vacuum, but also when they move through metal-semiconductor or metal-dielectric contacts.

6. Currents in vacuum limited by space charge. The Law of Three Seconds

At high current densities of thermionic emission, the current-voltage characteristic is significantly affected by the volume negative charge that arises between the cathode and anode. This volume negative charge prevents the electrons emitted from the cathode from reaching the anode. Thus, the anode current is less than the electron emission current from the cathode. When a positive potential is applied to the anode, the additional potential barrier at the cathode created by the space charge decreases and the anode current increases. This is a qualitative picture of the influence of the space charge on the current-voltage characteristic of the thermal diode. Theoretically, this question was investigated by Langmuir in 1913.

Under a number of simplifying assumptions, we calculate the dependence of the thermal diode current on the external potential difference applied between the anode and cathode and find the distribution of the field, potential, and electron concentration between the anode and cathode, taking into account the space charge.

Rice. 14. To the conclusion of the law of "three second"

Assume that the electrodes of the diode are flat. With a small distance between the anode and cathode d they can be considered infinitely large. We place the origin of coordinates on the cathode surface, and the axis X direct it perpendicular to this surface towards the anode (Fig. 14). The temperature of the cathode will be maintained constant and equal to T. Electrostatic field potential j , existing in the space between the anode and the cathode, will be a function of only one coordinate X. He must satisfy Poisson equation

,(20)

Here r is the bulk charge density; n is the electron concentration; j , r And n are functions of the coordinate X.

Given that the current density between the cathode and anode

and the speed of the electron v can be determined from the equation

Where m is the mass of the electron, equation (20) can be converted to the form

, .(21)

This equation must be supplemented with boundary conditions

These boundary conditions follow from the fact that the potential and electric field strength at the cathode surface must vanish. Multiplying both sides of equation (21) by dj /dx, we get

.(23)

Given that

(24a)

And ,(24b)

we write (23) as

.(25)

Now we can integrate both parts of Eq. (25) over X ranging from 0 to that value x, at which the potential is j . Then, taking into account the boundary conditions (22), we obtain

Integrating both parts of (27) within X=0, j =0 to X=1, j= Va, we get

.(28)

Squaring both sides of equality (28) and expressing the current density j from A according to (21), we get

.(30)

Formula (29) is called Langmuir's "law of three second".

This law is valid for electrodes of arbitrary shape. The expression for the numerical coefficient depends on the shape of the electrodes. The formulas obtained above make it possible to calculate the distributions of the potential, electric field strength, and electron density in the space between the cathode and anode. Integration of expression (26) within the limits of X=0 until the value when the potential is j , leads to the relation

those. the potential changes in proportion to the distance from the cathode X to the power of 4/3. Derivative dj/ dx characterizes the electric field strength between the electrodes. According to (26), the magnitude of the electric field strength E ~X 19 . Finally, the electron concentration

(32)

and, according to (31) n(x)~ (1/x) 2/9 .

Dependencies j (X ), E(X) And n(X) are shown in fig. 15. If X→0, then the concentration tends to infinity. This is a consequence of neglecting the thermal velocities of electrons near the cathode. In a real situation with thermionic emission, electrons leave the cathode not with zero velocity, but with a certain finite emission velocity. In this case, the anode current will exist even if there is a small reverse electric field near the cathode. Consequently, the volume charge density can change to such values ​​at which the potential near the cathode decreases to negative values ​​(Fig. 16). As the anode voltage increases, the potential minimum decreases and approaches the cathode (curves 1 and 2 in Fig. 16). At a sufficiently high voltage at the anode, the potential minimum merges with the cathode, the field strength at the cathode becomes equal to zero, and the dependence j (X) approaches (29), calculated without taking into account the initial electron velocities (curve 3 in Fig. 16). At high anode voltages, the space charge is almost completely absorbed and the potential between the cathode and anode varies linearly (curve 4, Fig. 16).

Thus, the potential distribution in the interelectrode space, taking into account the initial electron velocities, differs significantly from that which is the basis of the idealized model when deriving the "three second" law. This leads to a change and dependence of the anode current density. The calculation that takes into account the initial electron velocities for the case of the potential distribution shown in Fig. 17, and for cylindrical electrodes gives the following dependence for the total current of thermionic emission I (I=jS, Where S is the cross-sectional area of ​​the thermal current ):

.(33)

Options x m And Vm determined by the type of dependence j (X), their meaning is clear from Fig. 17. Parameter X m is equal to the distance from the cathode at which the potential reaches its minimum value = Vm. Factor C(x m), except x m, depends on the cathode and anode radii. Equation (33) is valid for small changes in the anode voltage, since And X m And Vm, as discussed above, depend on the anode voltage.

Thus, the law of "three second" is not universal, it is valid only in a relatively narrow range of voltages and currents. However, it is a clear example of a non-linear relationship between the current and voltage of an electronic device. The non-linearity of the current-voltage characteristic is the most important feature of many elements of radio and electrical circuits, including elements of solid state electronics.

Part 2. Laboratory work

7. Experimental setup for studying thermionic emission

Laboratory works No. 1 and 2 are performed on the same laboratory setup, implemented on the basis of a universal laboratory bench. The setup diagram is shown in fig. 18. In the measuring section there is a vacuum diode EL with a cathode of direct or indirect heating. The contacts of the filament "Incandescence", the anode "Anode" and the cathode "Cathode" are brought to the front panel of the measuring section. The source of heat is a stabilized direct current source type B5-44A. The I icon in the diagram indicates that the source is operating in the current stabilization mode. The procedure for working with a direct current source can be found in the technical description and operating instructions for this device. Similar descriptions are available for all electrical measuring instruments used in laboratory work. The anode circuit includes a stabilized DC source B5-45A and a universal digital voltmeter V7-21A, used in the DC measurement mode to measure the anode current of the thermal diode. To measure the anode voltage and the cathode filament current, you can use the devices built into the power source, or connect an additional RV7-32 voltmeter for a more accurate measurement of the voltage at the cathode.

The measuring section may contain vacuum diodes with different working cathode filaments. At the rated filament current, the diode operates in the mode of limiting the anode current by the space charge. This mode is required to complete Lab #1. Laboratory work No. 2 is performed at reduced filament currents, when the influence of the space charge is insignificant. When setting the filament current, you should be especially careful, because. the excess of the filament current over its nominal value for a given electron tube leads to a burnout of the cathode filament and the diode is out of order. Therefore, when preparing for work, be sure to check with the teacher or engineer the value of the working current of the glow of the diode used in the work, be sure to write down the data in a workbook and use it when compiling a report on laboratory work.

8. Laboratory work No. 1. Study of the effect of space charge on volt-amperethermal current characteristic

The purpose of the work: experimental study of the dependence of thermionic emission current on the anode voltage, determination of the exponent in the law of "three second".

Volt-ampere The characteristic of the thermionic emission current is described by the “three-seconds” law (see Section 6). This mode of operation of the diode occurs at sufficiently high cathode filament currents. Typically, at rated filament current, the vacuum diode current is limited by the space charge.

The experimental setup for performing this laboratory work is described in Sec. 7. In work, it is necessary to take the current-voltage characteristic of the diode at the rated filament current. The value of the operating current on the scale of the vacuum tube used should be taken from a teacher or engineer and written down in a workbook.

Work order

1. Familiarize yourself with the description and procedure for working with the devices necessary for the operation of the experimental setup. Assemble the circuit according to Fig. 18. The installation can be connected to the network only after checking the correctness of the assembled circuit by an engineer or teacher.

2. Turn on the power source of the cathode filament current and set the required filament current. Since when the filament current changes, the temperature and resistance of the filament change, which, in turn, leads to a change in the filament current, the adjustment must be carried out by the method of successive approximations. After the end of the adjustment, it is necessary to wait approximately 5 minutes for the filament current and the cathode temperature to stabilize.

3. Connect a constant voltage source to the anode circuit and, by changing the voltage at the anode, take the current-voltage characteristic point by point. Take the current-voltage characteristic in the range of 0 ... 25 V, every 0.5 ... 1 V.

I a(Va), Where I a– anode current, Va is the anode voltage.

5. If the anode voltage variation range is taken small, then the values x m, C(x,n) And Vm, included in formula (33), can be taken constant. At large Va magnitude Vm can be neglected. As a result, formula (33) is transformed into the form (after passing from the thermal current density j to its full value I)

6. From formula (34) determine the value WITH for the three maximum values ​​of the anode voltage on the current-voltage characteristic. Calculate the arithmetic mean of the obtained values. Substituting this value into formula (33), determine the value Vm for the three minimum anode voltages and calculate the arithmetic mean Vm.

7. Using the received value Vm, plot ln I a from ln( Va+|Vm|). Determine the degree of dependence by the tangent of the angle of this graph I a(V a + Vm). It should be close to 1.5.

8. Issue a report on the work.

Report Requirements

5. Conclusions on the work.

Control questions

1. What is the phenomenon of thermionic emission called? Define the work function of an electron. What is the difference between thermodynamic and external work function?

2. Explain the reasons for the appearance of a potential barrier at the solid-vacuum interface.

3. Explain, based on the energy scheme of the metal and the electron energy distribution curve, the thermal emission of electrons from the metal.

4. Under what conditions is thermionic current observed? How can thermionic current be observed? How does the current of a thermal diode depend on the applied electric field?

5. State the law Richardson-Deshman

6. Explain the qualitative picture of the influence of a volumetric negative charge on the current-voltage characteristic of a thermal diode. Formulate Langmuir's Law of Three Seconds.

7. What are the distributions of the potential, electric field strength and electron density in the space between the cathode and anode at currents limited by the space charge?

8. What is the dependence of the thermal emission current on the voltage between the anode and cathode, taking into account the space charge and initial electron velocities? Explain the meaning of the parameters that define this dependence;

9. Explain the layout of the experimental setup for studying thermionic emission. Explain the purpose of the individual circuit elements.

10. Explain the method of experimental determination of the exponent in the law of "three-seconds".

9. Laboratory work No. 2. Study of thermionic emission at low emission current densities

The purpose of the work: to study the current-voltage characteristics of a thermal diode at a low cathode heating current. Determination from the experimental results of the contact potential difference between the cathode and the anode, the temperature of the cathode.

At low thermal current densities volt-ampere the characteristic has a characteristic form with an inflection point corresponding to the modulus of the contact potential difference between the cathode and the anode (Fig. 10). The cathode temperature can be determined as follows. Let us pass in equation (12) describing the volt-ampere characteristic of thermionic emission at low current densities, from the density of thermocurrent j to its full value I(j=I /S, Where S is the cross-sectional area of ​​the thermal current ). Then we get

Where I S is the saturation current.

Logarithmizing (35), we have

.(36)

Since equation (36) describes the current-voltage characteristic in the section to the left of the inflection point, then to determine the cathode temperature, it is necessary to take any two points in this section with anode currents I a 1, I a 2 and anode voltages U a 1, U a 2 respectively. Then, according to equation (36),

Hence, for the cathode temperature, we obtain the working formula

.(37)

Work order

To perform laboratory work, you must:

1. Familiarize yourself with the description and procedure for working with the devices necessary for the operation of the experimental setup. Assemble the circuit according to fig. 18. The installation can be connected to the network only after checking the correctness of the assembled circuit by an engineer or teacher.

2. Turn on the cathode filament current power supply and set the required filament current. After setting the current, it is necessary to wait approximately 5 minutes for the filament current and cathode temperature to stabilize.

3. Connect a constant voltage source to the anode circuit and, by changing the voltage at the anode, take the current-voltage characteristic point by point. Volt-ampere take a characteristic in the range of 0 ... 5 V. every 0.05 ... 0.2 V.

4. Present the measurement results on the graph in coordinates ln I a(Va), Where I a– anode current, Va is the anode voltage. Since in this work the contact potential difference is determined by a graphical method, the scale along the horizontal axis should be chosen in such a way that the accuracy of determination V K.R.P was not less than 0.1 V.

5. Based on the inflection point of the current-voltage characteristic, determine the contact potential difference between the anode and cathode.

6. Determine the cathode temperature for three pairs of points on the inclined linear section of the current-voltage characteristic to the left of the inflection point. The cathode temperature should be calculated using formula (37). Calculate the average temperature value from this data.

7. Prepare a report on the work.

Report Requirements

The report is drawn up on a standard sheet of A4 paper and must contain:

1. Basic information on the theory.

2. Scheme of the experimental setup and its brief description.

3. Results of measurements and calculations.

4. Analysis of the obtained experimental results.

5. Conclusions on the work.

Control questions

1. List the types of electron emission. What is the cause of the release of electrons in each type of electron emission?

2. Explain the phenomenon of thermionic emission. Define the work function of an electron from a solid. How can one explain the existence of a potential barrier at the solid-vacuum interface?

3. Explain, based on the energy scheme of the metal and the electron energy distribution curve, the thermal emission of electrons from the metal.

4. State the law Richardson-Deshman. Explain the physical meaning of the quantities included in this law.

5. What are the features of the current-voltage characteristic of the thermionic cathode at low emission current densities? How does the contact potential difference between the cathode and anode affect it?

6. What is the Schottky effect? How is this effect explained?

7. Explain the decrease in the potential barrier for electrons under the influence of an electric field.

8. How will the cathode temperature be determined in this lab?

9. Explain the method for determining the contact potential difference in this work.

10. Explain the scheme and purpose of individual elements of the laboratory setup.