Work done by the electrostatic field force when moving a charge
Potential nature of field forces.
Circulation of the tension vector
Consider the electrostatic field created by charge q. Let a test charge q0 move in it. At any point in the field, the charge q0 is acted upon by a force
where is the magnitude of the force, is the ort of the radius vector that determines the position of the charge q0 relative to the charge q. Since the force changes from point to point, we write the work of the electrostatic field force as the work of a variable force:
Due to the fact that we considered the movement of a charge from point 1 to point 2 along an arbitrary trajectory, we can conclude that the work of moving a point charge in an electrostatic field does not depend on the shape of the path, but is determined only by the initial and final position of the charge. This indicates that the electrostatic field is potential, and the Coulomb force is a conservative force. The work done to move a charge in such a field along a closed path is always zero.
Projection on the direction of the contour?.
Let us take into account that the work along a closed path is zero
CIRCULATION of the tension vector.
The circulation of the electrostatic field strength vector, taken along an arbitrary closed contour, is always equal to zero.
Potential.
The relationship between tension and potential.
Potential gradient.
Equipotential surfaces
Since the electrostatic field is potential, the work of moving a charge in such a field can be represented as the difference in the potential energies of the charge at the initial and final points of the path. (Work is equal to the decrease in potential energy, or the change in potential energy taken with a minus sign.)
The constant is determined from the condition that when the charge q0 is removed to infinity, its potential energy must be equal to zero.
Different test charges q0i placed at a given point in the field will have different potential energies at this point:
The ratio of Wpot i to the value of the test charge q0i placed at a given point in the field is a constant value for a given point in the field for all test charges. This relationship is called POTENTIAL.
POTENTIAL - energy characteristic of the electric field. POTENTIAL is numerically equal to the potential energy possessed by a unit positive charge at a given point in the field.
The work of moving a charge can be represented as
Potential is measured in Volts
EQUIPOTENTIAL SURFACES are called surfaces of equal potential (t = const). The work done to move a charge along an equipotential surface is zero.
The connection between voltage and potential q can be found based on the fact that the work done to move charge q on an elementary segment d? can be represented as
Potential gradient.
The field strength is equal to the potential gradient taken with a minus sign.
The potential gradient shows how the potential changes per unit length. The gradient is perpendicular to the function and directed in the direction of increasing function. Consequently, the tension vector is perpendicular to the equipotential surface and directed in the direction of decreasing potential.
Let's consider the field created by a system of N point charges q1, q2, ... qN. The distances from the charges to a given field point are equal to r1, r2, … rN. The work done by the forces of this field on the charge q0 will be equal to the algebraic sum of the work done by the forces of each charge separately.
Field potential generated by the system charges is defined as the algebraic sum of the potentials created at the same point by each charge separately.
Calculation of the potential difference of a plane, two planes, a sphere, a ball, a cylinder
Using the connection between q and we determine the potential difference between two arbitrary points
The potential difference of the field of a uniformly charged infinite plane with surface charge density y.
§ 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 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.
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 – various characteristics 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
Lecture by A.P. Zubarev
The work of field forces to move a charge.
Potential and potential difference of the electric field.
As follows from Coulomb’s law, the force acting on a point charge q in electric field, created by other charges, is central. Let us recall that the central force is the force whose line of action is directed along the radius vector connecting some fixed point O (the center of the field) with any point on the trajectory. From "Mechanics" it is known that everything central forces are potential. The work of these forces does not depend on the shape of the path of movement of the body on which they act, and is equal to zero along any closed contour (path of movement). As applied to the electrostatic field (see figure) below:
.
Drawing. To determine the work of electrostatic field forces.
That is, the work of field forces to move a charge q from point 1 to point 2 is equal in magnitude and opposite in sign to the work to move a charge from point 2 to point 1, regardless of the shape of the path of movement. Consequently, the work of field forces to move a charge can be represented by the difference in the potential energies of the charge at the initial and final points of the movement path:
Let's introduce potential electrostatic field φ, specifying it as a ratio:
, (dimension in SI: ).
Then the work of field forces to move a point charge q from point 1 to point 2 will be:
The potential difference is called electrical voltage. The dimension of voltage, like potential, is [U] = B.
It is believed that there are no electric fields at infinity, which means . This allows you to give determination of potential as the work that needs to be done to move the charge q = +1 from infinity to a given point in space. Thus, the electric field potential is its energy characteristics.
Relationship between electric field strength and potential. Potential gradient. Electric field circulation theorem.
Tension and potential are two characteristics of the same object - the electric field, therefore there must be a functional connection between them. Indeed, the work of field forces to move a charge q from one point in space to another can be represented in two ways:
Whence it follows that 
This is the desired connection between the intensity and potential of the electric field in differential form.
- a vector directed from a point with less potential to a point with greater potential (see figure below).
 |
Drawing. Vectors and gradφ.
In this case, the modulus of the tension vector is equal to
From the property of potentiality of the electrostatic field it follows that the work of field forces along a closed loop (φ 1 = φ 2) is equal to zero:
so we can write
The last equality reflects the essence of the second fundamental theorem of electrostatics - electric field circulation theorems, according to which the field circulation along an arbitrary closed contour is zero. This theorem is a direct consequence potentiality electrostatic field.
Equipotential lines and surfaces and their properties.
Lines and surfaces, all points of which have the same potential, are called equipotential. Their properties directly follow from the representation of the work of field forces and are illustrated in the figure:
 |
Drawing. Illustration of the properties of equipotential lines and surfaces.
1) 
- the work done to move a charge along an equipotential line (surface) is zero, because .
Electrostatic field- email field of a stationary charge.
Fel, acting on the charge, moves it, performing work.
In a uniform electric field Fel = qE is a constant value
Work field (el. force) does not depend on the shape of the trajectory and on a closed trajectory = zero.
Electrostatics(from electro... and static) , a branch of the theory of electricity that studies the interaction of stationary electric charges. It is carried out through an electrostatic field. The fundamental law of E. - Coulomb is the law that determines the force of interaction of stationary point charges depending on their size and the distance between them.
Electric charges are sources of electrostatic fields. This fact is expressed by Gauss's theorem. The electrostatic field is potential, that is, the work of the forces acting on the charge from the electrostatic field does not depend on the shape of the path.
The electrostatic field satisfies the equations:
div D= 4pr, rot E = 0,
Where D- vector of electrical induction (see Electrical and magnetic induction), E - electrostatic field strength, r - electric charge density. The first equation is the differential form of Gauss's theorem, and the second expresses the potential nature of the electrostatic field. These equations can be obtained as special case Maxwell's equations.
Typical problems of electronics are finding the distribution of charges on the surfaces of conductors based on the known total charges or potentials of each of them, as well as calculating the energy of a system of conductors based on their charges and potentials.
To establish a connection between the force characteristic of the electric field - tension and its energy characteristics - potential Let's consider the elementary work of electric field forces on an infinitesimal displacement of a point charge q:d A = qE d l, the same work is equal to the decrease in the potential energy of the charge q:d A = d W P = q d, where d is the change in the electric field potential over the travel length d l. Equating the right-hand sides of the expressions, we get: E d l d or in the Cartesian coordinate system
E x d x + E y d y + Ez d z =d , (1.8)
Where E x,E y,E z- projections of the tension vector on the axes of the coordinate system. Since expression (1.8) is a total differential, then for the projections of the intensity vector we have
Equipotential surface- a concept applicable to any potential vector field, for example, a static electric field or a Newtonian gravitational field (Gravity). An equipotential surface is a surface on which the scalar potential of a given potential field takes on a constant value. Another, equivalent, definition is a surface that is orthogonal to the field lines at any point.
The surface of a conductor in electrostatics is an equipotential surface. In addition, placing a conductor on an equipotential surface does not change the configuration of the electrostatic field. This fact is used in the image method, which allows the calculation of the electrostatic field for complex configurations.
In a gravitational field, the level of a stationary fluid is established along the equipotential surface. In particular, the level of the oceans passes along the equipotential surface of the Earth's gravitational field. The equipotential surface of the ocean level, extended to the surface of the Earth, is called the geoid and plays an important role in geodesy.
5.Electrical capacity- characteristic of a conductor, a measure of its ability to accumulate electrical charge. In electrical circuit theory, capacitance is the mutual capacitance between two conductors; capacitive element parameter electrical diagram, presented in the form of a two-terminal network. Such capacitance is defined as the ratio of the magnitude of the electric charge to the potential difference between these conductors.
In the SI system, capacitance is measured in farads. In the GHS system in centimeters.
For a single conductor, capacitance is equal to the ratio of the conductor's charge to its potential, assuming that all other conductors are at infinity and that the potential of the point at infinity is assumed to be zero. IN mathematical form this definition looks like
Where Q- charge, U- conductor potential.
Capacity is determined geometric dimensions and the shape of the conductor and the electrical properties of the environment (its dielectric constant) and does not depend on the material of the conductor. For example, the capacity of a conducting ball of radius R equal (in SI system):
C= 4πε 0 ε R.
The concept of capacitance also refers to a system of conductors, in particular, to a system of two conductors separated by a dielectric - a capacitor. In this case mutual capacitance of these conductors (capacitor plates) will be equal to the ratio of the charge accumulated by the capacitor to the potential difference between the plates. For a parallel plate capacitor the capacitance is equal to:
Where S- area of one plate (it is assumed that they are equal), d- distance between the plates, ε - relative dielectric constant of the medium between the plates, ε 0 = 8.854×10 −12 F/m - electrical constant.
At parallel connection k capacitors, the total capacitance is equal to the sum of the capacitances of the individual capacitors:
C = C 1+ C 2+ … + C k .
At serial connection k capacitors, the reciprocal values of the capacitances are added:
1/C = 1/C 1+ 1/C 2+ … + 1/C k .
The energy of the electric field of a charged capacitor is equal to:
W = qU / 2 = CU 2 /2 = q 2/ (2C).
6.Electric current is calledpermanent , if the current strength and its direction do not change over time.
Current strength (often simply " current") in Explorer - scalar quantity, numerically equal to the charge flowing per unit time through the cross section of the conductor. Denoted by a letter (in some courses - . Not to be confused with vector current density):
The basic formula used to solve problems is Ohm's Law:
§ for a section of an electrical circuit:
Current is equal to the ratio of voltage to resistance.
§ for a complete electrical circuit:
Where E is the emf, R is the external resistance, r is the internal resistance.
The SI unit is 1 Ampere (A) = 1 Coulomb/second.
To measure current, a special device is used - an ammeter (for devices designed to measure small currents, the names milliammeter, microammeter, galvanometer are also used). It is included in the open circuit in the place where the current strength needs to be measured. The main methods for measuring current strength are: magnetoelectric, electromagnetic and indirect (by measuring voltage at a known resistance with a voltmeter).
When alternating current distinguish between instantaneous current strength, amplitude (peak) current strength and effective current strength (equal to the strength of direct current, which releases the same power).
Current density - a vector physical quantity that has the meaning of current flowing through a unit area. For example, with a uniform density distribution:
Current over the cross section of the conductor.
Among the conditions necessary for the existence of electric current are:
presence of free electric charges in the environment
· creation of an electric field in the environment
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.
Electromotive force (emf), a physical quantity characterizing the action of third-party (non-potential) forces in direct or alternating current sources; in a closed conducting circuit is equal to the work of these forces to move a single positive charge along the circuit. If through E p to indicate the field strength of external forces, then the emf in closed loop (L) is equal to , Where dl- contour length element.
The potential forces of an electrostatic (or stationary) field cannot support D.C. in the circuit, since the work of these forces on a closed path is zero. The passage of current through the conductors is accompanied by the release of energy - heating of the conductors. Third-party forces set in motion charged particles inside current sources: generators, galvanic cells, batteries, etc. The origin of third-party forces can be different. In generators, third-party forces are forces from the vortex electric field that arises when the magnetic field changes over time, or the Lorentz force acting from the magnetic field on electrons in a moving conductor; in galvanic cells and batteries - these are chemical forces, etc. Emf determines the current strength in the circuit at a given resistance (see Ohm's law) . EMF, like voltage, is measured in volts.
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 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. During treatment with an electrostatic shower on electrodes electric machine a potential difference of 100 kV is applied. 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
According to Einthoven's theory, which underlies electrocardiography, the heart is an electric dipole located in the center equilateral triangle(Einthoven triangle), the vertices of which can conventionally be considered 
located in right hand, left arm 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 of the body (in in this case between the vertices of Einthoven's triangle) is called abduction.
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.
- 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's put two in contact various metal 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 it when the work of moving the electron due to contact difference potentials will become equal to the difference in 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.
- 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 ahead of 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 functional state 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 most heat– in the armpit area, 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 midline bodies. 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 varies depending on different sides membranes, 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 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 electromagnetic wave via 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.
Because the electric currents interact with each other through their magnetic fields, the quantitative characteristics 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. 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.
