Electrostatic force formula. Basic formulas of electrostatics. Potential energy of interaction of charges

Cheat sheet with formulas in physics for the exam

and not only (may need 7, 8, 9, 10 and 11 classes).

For starters, a picture that can be printed in a compact form.

Mechanics

Pressure P=F/S
Density ρ=m/V
Pressure at the depth of the liquid P=ρ∙g∙h
Gravity Ft=mg
5. Archimedean force Fa=ρ w ∙g∙Vt
Equation of motion for uniformly accelerated motion

X=X0 + υ 0∙t+(a∙t 2)/2 S=( υ 2 -υ 0 2) /2а S=( υ +υ 0) ∙t /2

Velocity equation for uniformly accelerated motion υ =υ 0 +a∙t
Acceleration a=( υ -υ 0)/t
Circular speed υ =2πR/T
Centripetal acceleration a= υ 2/R
Relationship between period and frequency ν=1/T=ω/2π
Newton's II law F=ma
Hooke's law Fy=-kx
Law of universal gravitation F=G∙M∙m/R 2
The weight of a body moving with acceleration a P \u003d m (g + a)
The weight of a body moving with acceleration a ↓ P \u003d m (g-a)
Friction force Ffr=µN
Body momentum p=m υ
Force impulse Ft=∆p
Moment M=F∙ℓ
Potential energy body raised above the ground Ep=mgh
Potential energy of elastically deformed body Ep=kx 2 /2
Kinetic energy of the body Ek=m υ 2 /2
Work A=F∙S∙cosα
Power N=A/t=F∙ υ
Efficiency η=Ap/Az
Oscillation period of the mathematical pendulum T=2π√ℓ/g
Oscillation period spring pendulum T=2π √m/k
The equation of harmonic oscillations Х=Хmax∙cos ωt
Relationship of the wavelength, its speed and period λ= υ T

Molecular physics and thermodynamics

Amount of substance ν=N/ Na
Molar mass M=m/ν
Wed. kin. energy of monatomic gas molecules Ek=3/2∙kT
Basic equation of MKT P=nkT=1/3nm 0 υ 2
Gay-Lussac law (isobaric process) V/T =const
Charles' law (isochoric process) P/T =const
Relative humidity φ=P/P 0 ∙100%
Int. ideal energy. monatomic gas U=3/2∙M/µ∙RT
Gas work A=P∙ΔV
Boyle's law - Mariotte (isothermal process) PV=const
The amount of heat during heating Q \u003d Cm (T 2 -T 1)
The amount of heat during melting Q=λm
The amount of heat during vaporization Q=Lm
The amount of heat during fuel combustion Q=qm
State equation ideal gas PV=m/M∙RT
First law of thermodynamics ΔU=A+Q
Efficiency of heat engines η= (Q 1 - Q 2) / Q 1
Ideal efficiency. engines (Carnot cycle) η \u003d (T 1 - T 2) / T 1

Electrostatics and electrodynamics - formulas in physics

Coulomb's law F=k∙q 1 ∙q 2 /R 2
tension electric field E=F/q
Email tension. field of a point charge E=k∙q/R 2
Surface charge density σ = q/S
Email tension. fields of the infinite plane E=2πkσ
Dielectric constant ε=E 0 /E
Potential energy of interaction. charges W= k∙q 1 q 2 /R
Potential φ=W/q
Point charge potential φ=k∙q/R
Voltage U=A/q
For a uniform electric field U=E∙d
Electric capacity C=q/U
Capacitance of a flat capacitor C=S∙ ε ∙ε 0/d
Energy of a charged capacitor W=qU/2=q²/2С=CU²/2
Current I=q/t
Conductor resistance R=ρ∙ℓ/S
Ohm's law for the circuit section I=U/R
The laws of the last compounds I 1 \u003d I 2 \u003d I, U 1 + U 2 \u003d U, R 1 + R 2 \u003d R
Parallel laws. conn. U 1 \u003d U 2 \u003d U, I 1 + I 2 \u003d I, 1 / R 1 + 1 / R 2 \u003d 1 / R
Electric current power P=I∙U
Joule-Lenz law Q=I 2 Rt
Ohm's law for complete chain I=ε/(R+r)
Short circuit current (R=0) I=ε/r
Magnetic induction vector B=Fmax/ℓ∙I
Ampere Force Fa=IBℓsin α
Lorentz force Fл=Bqυsin α
Magnetic flux Ф=BSсos α Ф=LI
Law electromagnetic induction Ei=ΔF/Δt
EMF of induction in moving conductor Ei=Вℓ υ sinα
EMF of self-induction Esi=-L∙ΔI/Δt
Energy magnetic field coils Wm=LI 2 /2
Oscillation period count. contour T=2π ∙√LC
Inductive reactance X L =ωL=2πLν
Capacitance Xc=1/ωC
The current value of the current Id \u003d Imax / √2,
RMS voltage Ud=Umax/√2
Impedance Z=√(Xc-X L) 2 +R 2

Optics

The law of refraction of light n 21 \u003d n 2 / n 1 \u003d υ 1 / υ 2
Refractive index n 21 =sin α/sin γ
Thin lens formula 1/F=1/d + 1/f
Optical power of the lens D=1/F
max interference: Δd=kλ,
min interference: Δd=(2k+1)λ/2
Differential grating d∙sin φ=k λ

The quantum physics

Einstein's formula for the photoelectric effect hν=Aout+Ek, Ek=U ze
Red border of the photoelectric effect ν to = Aout/h
Photon momentum P=mc=h/ λ=E/s

Physics of the atomic nucleus

Encyclopedic YouTube

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The works of Coulomb laid the foundation for electrostatics (although ten years before him, Cavendish obtained the same results, even with even greater accuracy. The results of Cavendish's work were kept in the family archive and were published only a hundred years later); the law of electrical interactions found by the latter made it possible for Green, Gauss and Poisson to create a mathematically elegant theory. The most essential part of electrostatics is the theory of potential created by Green and Gauss. A great deal of experimental research on electrostatics was carried out by Rees, whose books were in former times the main aid in the study of these phenomena.

The dielectric constant

Finding the value of the dielectric coefficient K of any substance, a coefficient included in almost all the formulas that have to be dealt with in electrostatics, can be done in very different ways. The most commonly used methods are as follows.
1) Comparison of electric capacitances of two capacitors having the same dimensions and shape, but in which one has an insulating layer of air, the other has a layer of the tested dielectric.
2) Comparison of attraction between the surfaces of the capacitor, when these surfaces are given a certain potential difference, but in one case there is air between them (attractive force \u003d F 0), in the other case - the test liquid insulator (attractive force \u003d F). The dielectric coefficient is found by the formula:
K = F 0 F . (\displaystyle K=(\frac (F_(0))(F)).)
3) Observations of electric waves (see Electrical oscillations) propagating along wires. According to Maxwell's theory, the propagation velocity of electric waves along the wires is expressed by the formula
V = 1 K μ . (\displaystyle V=(\frac (1)(\sqrt (K\mu ))).)
in which K denotes the dielectric coefficient of the medium surrounding the wire, μ denotes the magnetic permeability of this medium. It is possible to set μ = 1 for the vast majority of bodies, and therefore it turns out
V = 1 K . (\displaystyle V=(\frac (1)(\sqrt (K))).)
Usually, the lengths of standing electric waves arising in parts of the same wire in air and in the tested dielectric (liquid) are usually compared. Having determined these lengths λ 0 and λ, we get K = λ 0 2 / λ 2. According to Maxwell's theory, it follows that when an electric field is excited in any insulating substance, special deformations occur inside this substance. Along the induction tubes, the insulating medium is polarized. Electric displacements arise in it, which can be likened to displacements of positive electricity in the direction of the axes of these tubes, and through each transverse section The amount of electricity passing through the tube is equal to
D = 1 4 π K F . (\displaystyle D=(\frac (1)(4\pi ))KF.)
Maxwell's theory makes it possible to find expressions for those internal forces (forces of tension and pressure) that appear in dielectrics when an electric field is excited in them. This question was first considered by Maxwell himself, and later and more thoroughly by Helmholtz. Further development of the theory of this issue and the theory of electrostriction (that is, a theory that considers phenomena that depend on the occurrence of special voltages in dielectrics when an electric field is excited in them) belongs to the works of Lorberg, Kirchhoff, P. Duhem, N. N. Schiller and some others.

Border conditions

Let's finish summary most significant of the department of electrostriction by considering the question of the refraction of induction tubes. Imagine two dielectrics in an electric field, separated from each other by some surface S, with dielectric coefficients K 1 and K 2 .
Let at the points P 1 and P 2 located infinitely close to the surface S on either side, the magnitudes of the potentials are expressed through V 1 and V 2, and the magnitude of the forces experienced by the unit of positive electricity placed at these points through F 1 and F 2. Then for a point P lying on the surface S itself, it should be V 1 = V 2,
d V 1 d s = d V 2 d s , (30) (\displaystyle (\frac (dV_(1))(ds))=(\frac (dV_(2))(ds)),\qquad (30))
if ds represents an infinitesimal displacement along the line of intersection of the tangent plane to the surface S at point P with the plane passing through the normal to the surface at that point and through the direction of the electric force at it. On the other hand, it should be
K 1 d V 1 d n 1 + K 2 d V 2 d n 2 = 0. (31) (\displaystyle K_(1)(\frac (dV_(1))(dn_(1)))+K_(2)( \frac (dV_(2))(dn_(2)))=0.\qquad (31))
Denote by ε 2 the angle formed by the force F2 with the normal n2 (inside the second dielectric), and through ε 1 the angle formed by the force F 1 with the same normal n 2 Then, using formulas (31) and (30), we find
t g ε 1 t g ε 2 = K 1 K 2 . (\displaystyle (\frac (\mathrm (tg) (\varepsilon _(1)))(\mathrm (tg) (\varepsilon _(2))))=(\frac (K_(1))(K_( 2))).)
So, on the surface separating two dielectrics from each other, the electric force undergoes a change in its direction, like a light beam entering from one medium to another. This consequence of the theory is justified by experience.
Basic Laws of Dynamics. Newton's laws - first, second, third. Galileo's principle of relativity. The law of universal gravitation. Gravity. Forces of elasticity. The weight. Friction forces - rest, sliding, rolling + friction in liquids and gases.
Kinematics. Basic concepts. Uniform rectilinear motion. Uniform movement. Uniform circular motion. Reference system. Trajectory, displacement, path, equation of motion, speed, acceleration, relationship between linear and angular velocity.
simple mechanisms. Lever (lever of the first kind and lever of the second kind). Block (fixed block and movable block). Inclined plane. Hydraulic Press. The golden rule of mechanics
Conservation laws in mechanics. Mechanical work, power, energy, law of conservation of momentum, law of conservation of energy, equilibrium of solids
Circular movement. Equation of motion in a circle. Angular velocity. Normal = centripetal acceleration. Period, frequency of circulation (rotation). Relationship between linear and angular velocity
Mechanical vibrations. Free and forced vibrations. Harmonic vibrations. Elastic oscillations. Mathematical pendulum. Energy transformations during harmonic vibrations
mechanical waves. Velocity and wavelength. Traveling wave equation. Wave phenomena (diffraction, interference...)
Hydromechanics and Aeromechanics. Pressure, hydrostatic pressure. Pascal's law. Basic equation of hydrostatics. Communicating vessels. Law of Archimedes. Sailing conditions tel. Fluid flow. Bernoulli's law. Torricelli formula
Molecular physics. Basic provisions of the ICT. Basic concepts and formulas. Properties of an ideal gas. Basic equation of the MKT. Temperature. The equation of state for an ideal gas. Mendeleev-Klaiperon equation. Gas laws - isotherm, isobar, isochore
Wave optics. Corpuscular-wave theory of light. Wave properties of light. dispersion of light. Light interference. Huygens-Fresnel principle. Diffraction of light. Light polarization
Thermodynamics. Internal energy. Job. Quantity of heat. Thermal phenomena. First law of thermodynamics. Application of the first law of thermodynamics to various processes. Heat balance equation. The second law of thermodynamics. Heat engines
You are here now: Electrostatics. Basic concepts. Electric charge. The law of conservation of electric charge. Coulomb's law. The principle of superposition. The theory of close action. Electric field potential. Capacitor.
Constant electric current. Ohm's law for a circuit section. Operation and DC power. Joule-Lenz law. Ohm's law for a complete circuit. Faraday's law of electrolysis. Electrical circuits - serial and parallel connection. Kirchhoff's rules.
Electromagnetic vibrations. Free and forced electromagnetic oscillations. Oscillatory circuit. Alternating electric current. Capacitor in AC circuit. An inductor ("solenoid") in an alternating current circuit.
Elements of the theory of relativity. Postulates of the theory of relativity. Relativity of simultaneity, distances, time intervals. Relativistic law of addition of velocities. The dependence of mass on speed. The basic law of relativistic dynamics...
Errors of direct and indirect measurements. Absolute, relative error. Systematic and random errors. Standard deviation (error). Table for determining the errors of indirect measurements of various functions.

Electrostatic (or Coulomb) repulsion occurs between like-charged bodies, and electrostatic attraction between oppositely charged bodies. The phenomenon of repulsion of like charges underlies the creation of an electroscope - a device for detecting electric charges.

Electrostatics is based on Coulomb's law. This law describes the interaction of point electric charges.

The foundation of electrostatics was laid by the work of Coulomb (although ten years before him, Cavendish obtained the same results, even with even greater accuracy. The results of Cavendish's work were kept in the family archive and were published only a hundred years later); the law of electrical interactions found by the latter made it possible for Green, Gauss and Poisson to create a mathematically elegant theory. The most essential part of electrostatics is the potential theory created by Green and Gauss. A great deal of experimental research on electrostatics was carried out by Rees, whose books were in former times the main aid in the study of these phenomena.

Faraday's experiments, carried out as far back as the first half of the thirties of the 19th century, should have entailed a radical change in the basic provisions of the doctrine of electrical phenomena. These experiments showed that what was considered to be completely passive relating to electricity, namely, insulating substances or, as Faraday called them, dielectrics, is of decisive importance in all electrical processes and, in particular, in the very electrification of conductors. These experiments revealed that the substance of the insulating layer between the two surfaces of the capacitor plays important role in the capacitance of this capacitor. The replacement of air, as an insulating layer between the surfaces of the capacitor, by some other liquid or solid insulator, has the same effect on the value of the electrical capacity of the capacitor, which has a corresponding decrease in the distance between these surfaces while maintaining air as an insulator. When the air layer is replaced by a layer of another liquid or solid dielectric, the electrical capacitance of the capacitor increases by a factor of K. This value K is called by Faraday the inductive capacity of a given dielectric. Today, the value of K is usually called the dielectric constant of this insulating substance.

The same change in electrical capacitance occurs in each individual conducting body when this body is transferred from the air to another insulating medium. But a change in the electric capacitance of a body entails a change in the magnitude of the charge on this body at a given potential on it, and vice versa, a change in the potential of the body at a given charge. At the same time, it also changes the electrical energy of the body. So, the value of the insulating medium in which the electrified bodies are placed or which separates the surfaces of the capacitor is extremely significant. The insulating substance not only retains an electrical charge on the surface of the body, it affects the very electrical state of the latter. Such is the conclusion to which Faraday's experiments led. This conclusion was quite consistent with Faraday's basic view of electrical actions.

According to Coulomb's hypothesis, electrical actions between bodies were considered as actions that occur at a distance. It was assumed that two charges q and q ", mentally concentrated at two points separated from each other by a distance r, repel or attract one another along the direction of the line connecting these two points, with a force that is determined by the formula

Moreover, the coefficient C is dependent solely on the units used to measure the values of q, r and f. The nature of the medium inside which these two points with charges q and q "are located, was assumed to be of no importance, does not affect the value of f. Faraday held a completely different view of this. In his opinion, an electrified body only apparently acts on another body , located at a certain distance from it; in fact, the electrified body only causes special changes in the insulating medium in contact with it, which are transmitted in this medium from layer to layer, finally reach the layer immediately adjacent to the other body under consideration and produce there something , which appears as the direct action of the first body on the second through the medium separating them. With such a view of electrical actions, Coulomb's law, expressed by the above formula, can only serve to describe what observation gives, and does not in the least express the true process that takes place in this case. Then it becomes clear that, in general, electrical actions change with a change from olating medium, since in this case the deformations that arise in the space between two, apparently, electrified bodies acting on each other, must also change. Coulomb's law, so to speak, describing the phenomenon externally, must be replaced by another, which includes a characteristic of the nature of the insulating medium. For an isotropic and homogeneous medium, the Coulomb law, as shown by further studies, can be expressed by the following formula:

Here K denotes what is referred to above as the dielectric constant of a given insulating medium. The value of K for air is equal to unity, i.e. for air, the interaction between two points with charges q and q "is expressed as Coulomb accepted it.

According to the basic idea of Faraday, the surrounding insulating medium, or, better, those changes (polarization of the medium), which, under the influence of a process that brings bodies into an electrical state, occur in the ether filling this medium, are the cause of all electrical actions that we observe. According to Faraday, the very electrization of conductors on their surface is only a consequence of the influence of polarized energy on them. environment. In this case, the insulating medium is in a stressed state. Based on very simple experiments, Faraday came to the conclusion that when electric polarization is excited in any medium, when an electric field is excited, as they say now, in this medium there must be tension along the lines of force (a line of force is a line tangent to which coincide with the directions of the electric forces experienced by the positive electricity, imagined at the points located on this line) and there must be pressure in directions perpendicular to the lines of force. Such a stressed state can only be induced in insulators. The vehicles are incapable of experiencing such a change in their state, there is no perturbation in them; and only on the surface of such conducting bodies, i.e., on the boundary between the conductor and the insulator, does the polarized state of the insulating medium become noticeable, it is expressed in the apparent distribution of electricity on the surface of the conductors. So, the electrified conductor is, as it were, connected with the surrounding insulating medium. From the surface of this electrified conductor, as it were, lines of force, and these lines end on the surface of another conductor, which appears to be covered by an opposite sign of electricity. This is the picture that Faraday painted for himself to explain the phenomena of electrification.

Faraday's doctrine was not soon accepted by physicists. Faraday's experiments were considered even in the sixties, as not giving the right to assume any significant role of insulators in the processes of electrification of conductors. Only later, after the appearance of the remarkable works of Maxwell, Faraday's ideas began to spread more and more among scientists and, finally, were recognized as fully consistent with the facts.

It is appropriate to note here that back in the sixties, Prof. F. N. Shvedov, on the basis of his experiments, very ardently and convincingly proved the correctness of Faraday's main provisions regarding the role of insulators. In fact, however, many years before Faraday's work, the influence of insulators on electrical processes had already been discovered. Back in the early 70s of the 18th century, Cavendish observed and very carefully studied the significance of the nature of the insulating layer in a capacitor. Cavendish's experiments, as well as later Faraday's experiments, showed an increase in the capacitance of a capacitor when the air layer in this capacitor is replaced by a layer of some solid dielectric of the same thickness. These experiments even make it possible to determine the numerical values of the dielectric constants of some insulating substances, and these values turn out to be relatively slightly different from those found in recent times using more advanced measuring instruments. But this work of Cavendish, like his other studies on electricity, which led him to establish the law of electrical interactions, identical with the law published in 1785 by Coulomb, remained unknown until 1879. Only in this year, Cavendish's memoirs were published by Maxwell, who repeated almost all the experiments of Cavendish and who made many very valuable indications about them.

Potential

Assuming C = 1, i.e. when expressing the amount of electricity in the so-called absolute electrostatic unit of the CGS system, this Coulomb's law gets the expression:

Hence the potential function or, more simply, the potential at a point whose coordinates (x, y, z) is determined by the formula:

In which the integral extends to all electric charges in a given space, and r denotes the distance of the charge element dq to the point (x, y, z). Denoting the surface density of electricity on electrified bodies by σ, and the volumetric density of electricity in them by ρ, we have

Here dS denotes the body surface element, (ζ, η, ξ) are the coordinates of the body volume element. The projections on the coordinate axes of the electric force F experienced by a unit of positive electricity at the point (x, y, z) are found by the formulas:

Surfaces, at all points of which V = constant, are called equipotential surfaces or, more simply, level surfaces. The lines orthogonal to these surfaces are electrical lines of force. The space in which electric forces can be detected, i.e., in which lines of force can be built, is called the electric field. The force experienced by a unit of electricity at any point in this field is called the voltage of the electric field at that point. The function V has the following properties: it is single-valued, finite, and continuous. It can also be set to vanish at points that are infinitely far away from a given distribution of electricity. The potential remains the same value at all points of any conducting body. For all points the globe, as well as for all conductors metallically connected to the ground, the function V is equal to 0 (this does not pay attention to the Volta phenomenon, which was reported in the article Electrification). Denoting by F the magnitude of the electric force experienced by a unit of positive electricity at some point on the surface S that encloses a part of space, and by ε the angle formed by the direction of this force with the outward normal to the surface S at the same point, we have

In this formula, the integral extends to the entire surface S, and Q denotes the algebraic sum of the amount of electricity contained within the closed surface S. Equality (4) expresses a theorem known as the Gauss theorem. Simultaneously with Gauss, the same equality was obtained by Green, which is why some authors call this theorem Green's theorem. From the Gauss theorem can be deduced as corollaries,

here ρ denotes the volumetric density of electricity at the point (x, y, z);

this equation applies to all points where there is no electricity

Here Δ is the Laplace operator, n1 and n2 denote the normals at a point on some surface at which the surface density of electricity is σ, the normals drawn in either direction from the surface. It follows from the Poisson theorem that for a conducting body in which at all points V = constant, there must be ρ = 0. Therefore, the expression for the potential takes the form

From the formula expressing the boundary condition, i.e. from formula (7), it follows that on the surface of the conductor

Moreover, n denotes the normal to this surface, directed from the conductor into the insulating medium adjacent to this conductor. From the same formula, one derives

Here Fn denotes the force experienced by a unit of positive electricity located at a point infinitely close to the surface of the conductor, having at that place a surface density of electricity equal to σ. The force Fn is directed along the normal to the surface at this point. The force experienced by a unit of positive electricity, located in the electrical layer itself on the surface of the conductor and directed along the outer normal to this surface, is expressed through

Hence, the electric pressure experienced in the direction of the outer normal by each unit of the surface of the electrified conductor is expressed by the formula

The above equations and formulas make it possible to draw many conclusions related to the issues considered in E. But all of them can be replaced by even more general ones if we use what is contained in the theory of electrostatics given by Maxwell.

Maxwell electrostatics

Here the integral extends over any closed surface S, F denotes the magnitude of the electric force experienced by a unit of electricity at the center of the element of this surface dS, ε denotes the angle formed by this force with the outward normal to the surface element dS, K denotes the dielectric coefficient of the medium adjacent to element dS, and Q denotes the algebraic sum of the amounts of electricity contained within the surface S. The following equations are the consequences of expression (13):

These equations are more general than equations (5) and (7). They refer to the case of arbitrary isotropic insulating media. Function V, which is a general integral of equation (14) and satisfies at the same time equation (15) for any surface that separates two dielectric media with dielectric coefficients K 1 and K 2, as well as the condition V = constant. for each conductor in the electric field under consideration, is the potential at the point (x, y, z). It also follows from expression (13) that the apparent interaction of two charges q and q 1 located at two points located in a homogeneous isotropic dielectric medium at a distance r from each other can be represented by the formula

That is, this interaction is inversely proportional to the square of the distance, as it should be according to Coulomb's law. From equation (15) we get for the conductor:

These formulas are more general than the above (9), (10) and (12).

is an expression for the flow of electrical induction through the element dS. Having drawn through all points of the contour of the element dS lines coinciding with the directions F at these points, we obtain (for an isotropic dielectric medium) an induction tube. For all sections of such an induction tube, which does not contain electricity, it should be, as follows from equation (14),

KFCos ε dS = const.

It is not difficult to prove that if in any system of bodies the electric charges are in equilibrium when the densities of electricity are respectively σ1 and ρ1 or σ 2 and ρ 2, then the charges will be in equilibrium even when the densities are σ = σ 1 + σ 2 and ρ = ρ 1 + ρ 2  (the principle of addition of charges in equilibrium). It is equally easy to prove that under given conditions there can be only one distribution of electricity in the bodies that make up any system.

The property of a conducting closed surface, which is in connection with the earth, turns out to be very important. Such a closed surface is a screen, a protection for the entire space enclosed within it, from the influence of any electric charges located on the outer side of the surface. As a result, electrometers and other electrical measuring instruments are usually surrounded by metal cases connected to the ground. Experiments show that for such electric. screens, there is no need to use solid metal, it is quite enough to arrange these screens from metal meshes or even metal gratings.

A system of electrified bodies has energy, that is, it has the ability to perform a certain work with a complete loss of its electrical state. In electrostatics, the following expression is derived for the energy of a system of electrified bodies:

In this formula, Q and V denote, respectively, any amount of electricity in a given system and the potential at the place where this amount is located; the sign ∑ indicates that one should take the sum of the products VQ for all quantities Q of the given system. If the system of bodies is a system of conductors, then for each such conductor the potential has the same value at all points of this conductor, and therefore in this case the expression for energy takes the form:

Here 1, 2.. n are the icons of different conductors that are part of the system. This expression can be replaced by others, namely, the electrical energy of a system of conducting bodies can be represented either depending on the charges of these bodies, or depending on their potentials, i.e., the expressions can be applied to this energy:

In these expressions various coefficientsα and β depend on the parameters that determine the positions of conducting bodies in a given system, as well as their shapes and sizes. In this case, the coefficients β with two identical signs, such as β11, β22, β33, etc., represent the electric capacities (see Electric capacity) of bodies marked with these signs, the coefficients β with two different signs, such as β12, β23, β24, etc., are the coefficients of mutual induction of two bodies, the icons of which are next to this coefficient. Having the expression of electric energy, we obtain an expression for the force experienced by any body, whose icon is i, and from the action of which the parameter si, which serves to determine the position of this body, receives an increment. The expression of this force will be

Electrical energy can be represented in another way, namely, through

In this formula, the integration extends over the entire infinite space, F denotes the magnitude of the electric force experienced by a unit of positive electricity at the point (x, y, z), i.e. the electric field voltage at this point, and K denotes the dielectric coefficient at the same point . With such an expression for the electrical energy of a system of conducting bodies, this energy can be considered distributed only in insulating media, and the share of the element dxdyds of the dielectric accounts for the energy

Expression (26) fully corresponds to the views on electrical processes that were developed by Faraday and Maxwell.

In this formula, both triple integrals apply to the entire volume of any space A, double integrals - to all surfaces bounding this space, ∆V and ∆U denote the sums of the second derivatives of the functions V and U with respect to x, y, z; n is the normal to the element dS of the bounding surface directed inside the space A.

Examples

Example 1

how special case Green's formula yields a formula that expresses the above Gauss theorem. AT Encyclopedic Dictionary it is not appropriate to touch upon questions about the laws of distribution of electricity on various bodies. These questions are very difficult problems of mathematical physics, and various methods are used to solve such problems. We give here only for one body, namely, for an ellipsoid with semi-axes a, b, c, the expression for the surface density of electricity σ at the point (x, y, z). We find:

Here Q denotes the total amount of electricity that is on the surface of this ellipsoid. The potential of such an ellipsoid at some point on its surface, when there is a homogeneous isotropic insulating medium around the ellipsoid with a dielectric coefficient K, is expressed through

The electrical capacitance of the ellipsoid is obtained from the formula

Example 2

Using equation (14), assuming only ρ = 0 and K = constant in it, and formula (17), we can find an expression for the electric capacitance of a flat capacitor with a guard ring and a guard box, in which the insulating layer has a dielectric coefficient K. This is expression looks like

Here S denotes the value of the collecting surface of the capacitor, D is the thickness of its insulating layer. For a capacitor without a guard ring and a guard box, formula (28) will only give an approximate expression for the electric capacitance. For the electrical capacity of such a capacitor, the Kirchhoff formula is given. And even for a capacitor with a guard ring and a box, formula (29) does not represent a completely strict expression for the electric capacitance. Maxwell indicated the correction that should be made in this formula in order to obtain a more rigorous result.

The energy of a flat capacitor (with guard ring and box) is expressed in terms of

Here V1 and V2 are the potentials of the conducting surfaces of the capacitor.

Example 3

For a spherical capacitor, the expression for electric capacitance is obtained:

In which R 1 and R 2 denote, respectively, the radii of the inner and outer conductive surfaces of the capacitor. Using the expression for electric energy (formula 22), it is not difficult to establish the theory of absolute and quadrant electrometers

Finding the value of the dielectric coefficient K of any substance, a coefficient included in almost all the formulas that have to be dealt with in electrostatics, can be done in very different ways. The most commonly used methods are as follows.

1) Comparison of the capacitances of two capacitors having the same dimensions and shape, but in which one has an insulating layer of air, the other has a layer of the dielectric under test.

2) Comparison of attraction between the surfaces of a capacitor, when these surfaces are given a certain potential difference, but in one case there is air between them (attractive force \u003d F 0), in the other case - the test liquid insulator (attractive force \u003d F). The dielectric coefficient is found by the formula:

3) Observations of electric waves (see Electric oscillations) propagating along wires. According to Maxwell's theory, the propagation velocity of electric waves along the wires is expressed by the formula

In which K denotes the dielectric coefficient of the medium surrounding the wire, μ denotes the magnetic permeability of this medium. It is possible to set μ = 1 for the vast majority of bodies, and therefore it turns out

Usually, the lengths of standing electric waves arising in parts of the same wire in air and in the tested dielectric (liquid) are usually compared. Having determined these lengths λ 0 and λ, we get K = λ 0 2 / λ 2. According to Maxwell's theory, it follows that when an electric field is excited in any insulating substance, special deformations occur inside this substance. Along the induction tubes, the insulating medium is polarized. Electric displacements arise in it, which can be likened to the movements of positive electricity in the direction of the axes of these tubes, and through each cross section of the tube passes an amount of electricity equal to

Maxwell's theory makes it possible to find expressions for those internal forces (forces of tension and pressure) that appear in dielectrics when an electric field is excited in them. This question was first considered by Maxwell himself, and later and more thoroughly by Helmholtz. Further development of the theory of this issue and the theory of electrostriction closely connected with this (i.e., the theory that considers phenomena that depend on the occurrence of special voltages in dielectrics when an electric field is excited in them) belongs to the works of Lorberg, Kirchhoff, Duhem, N. N. Schiller and some others.

Border conditions

Let us conclude this summary of the most important of the department of electrostriction with a consideration of the question of the refraction of induction tubes. Imagine two dielectrics in an electric field, separated from each other by some surface S, with dielectric coefficients K 1 and K 2 . Let at the points P 1 and P 2 located infinitely close to the surface S on either side, the magnitudes of the potentials are expressed through V 1 and V 2, and the magnitude of the forces experienced by the unit of positive electricity placed at these points through F 1 and F 2. Then for a point P lying on the surface S itself, it should be V 1 = V 2,

if ds represents an infinitesimal displacement along the line of intersection of the tangent plane to the surface S at point P with the plane passing through the normal to the surface at that point and through the direction of the electric force at it. On the other hand, it should be

Let us denote by ε 2 the angle formed by the force F 2 with the normal n 2 (inside the second dielectric), and through ε 1 the angle formed by the force F 1 with the same normal n 2 Then, using formulas (31) and (30), we find

So, on the surface separating two dielectrics from each other, the electric force undergoes a change in its direction, like a light beam entering from one medium to another. This consequence of the theory is justified by experience.

From Wikipedia, the free encyclopedia

Portal:Physics

Electrostatics- a branch of the doctrine of electricity, studying the interaction of motionless electric charges.

Between of the same name charged bodies there is an electrostatic (or Coulomb) repulsion, and between differently charged - electrostatic attraction. The phenomenon of repulsion of like charges underlies the creation of an electroscope - a device for detecting electric charges.

Electrostatics is based on Coulomb's law. This law describes the interaction of point electric charges.

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The works of Coulomb laid the foundation for electrostatics (although ten years before him, Cavendish obtained the same results, even with even greater accuracy. The results of Cavendish's work were kept in the family archive and were published only a hundred years later); the law of electrical interactions found by the latter made it possible for Green, Gauss and Poisson to create a mathematically elegant theory. The most significant part of electrostatics is the theory of potential created by Green and Gauss. A great deal of experimental research on electrostatics was carried out by Rees, whose books were in former times the main aid in the study of these phenomena.

The dielectric constant

1) Comparison of electric capacitances of two capacitors having the same dimensions and shape, but in which one has an insulating layer of air, the other has a layer of the tested dielectric.

2) Comparison of attraction between the surfaces of the capacitor, when these surfaces are given a certain potential difference, but in one case there is air between them (attractive force \u003d F 0), in the other case - the test liquid insulator (attractive force \u003d F). The dielectric coefficient is found by the formula:

in which K denotes the dielectric coefficient of the medium surrounding the wire, μ denotes the magnetic permeability of this medium. It is possible to set μ = 1 for the vast majority of bodies, and therefore it turns out

Usually, the lengths of standing electric waves arising in parts of the same wire in air and in the tested dielectric (liquid) are usually compared. Having determined these lengths λ 0 and λ, we get K = λ 0 2 / λ 2. According to Maxwell's theory, it follows that when an electric field is excited in any insulating substance, special deformations occur inside this substance. Along the induction tubes, the insulating medium is polarized. Electric displacements arise in it, which can be likened to the movements of positive electricity in the direction of the axes of these tubes, and an amount of electricity passes through each cross section of the tube, equal to

Maxwell's theory makes it possible to find expressions for those internal forces (forces of tension and pressure) that appear in dielectrics when an electric field is excited in them. This question was first considered by Maxwell himself, and later and more thoroughly by Helmholtz. Further development of the theory of this issue and the theory of electrostriction (that is, a theory that considers phenomena that depend on the occurrence of special voltages in dielectrics when an electric field is excited in them) belongs to the works of Lorberg, Kirchhoff, P. Duhem, N. N. Schiller and some others.

Border conditions

Let at the points P 1 and P 2 located infinitely close to the surface S on either side, the magnitudes of the potentials are expressed through V 1 and V 2, and the magnitude of the forces experienced by the unit of positive electricity placed at these points through F 1 and F 2. Then for a point P lying on the surface S itself, it should be V 1 = V 2,

Denote by ε 2 the angle formed by the force F2 with the normal n2 (inside the second dielectric), and through ε 1 the angle formed by the force F 1 with the same normal n 2 Then, using formulas (31) and (30), we find

Literature

Landau, L. D., Lifshitz, E. M. Field theory. - Edition 7th, corrected. - M .: Nauka, 1988. - 512 p. - ("Theoretical Physics", Volume II). - ISBN 5-02-014420-7
Matveev A. N. electricity and magnetism. M.: graduate School, 1983.
Tunnel M.-A. Fundamentals of electromagnetism and the theory of relativity. Per. from fr. M.: Foreign Literature, 1962. 488 p.
Borgman, "Foundations of the doctrine of electrical and magnetic phenomena" (vol. I);
Maxwell, "Treatise on Electricity and Magnetism" (vol. I);
Poincaré, "Electricité et Optique"";
Wiedemann, "Die Lehre von der Elektricität" (vol. I);

Electrostatic force formula. Basic formulas of electrostatics. Potential energy of interaction of charges

Electrostatics and electrodynamics - formulas in physics

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