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Determination of the force of interaction of charges in a vacuum. Coulomb's law. Point charge

In 1785, the French physicist Charles Coulomb experimentally established the basic law of electrostatics - the law of interaction of two stationary point charged bodies or particles.

The law of interaction of stationary electric charges - Coulomb's law - is a basic (fundamental) physical law and can only be established experimentally. It does not follow from any other laws of nature.

If we denote the charge modules by | q 1 | and | q 2 |, then Coulomb’s law can be written in the following form:

\(~F = k \cdot \dfrac(|q_1| \cdot |q_2|)(r^2)\) , (1)

Where k– proportionality coefficient, the value of which depends on the choice of units of electric charge. In the SI system \(~k = \dfrac(1)(4 \pi \cdot \varepsilon_0) = 9 \cdot 10^9\) N m 2 / C 2, where ε 0 is the electrical constant equal to 8.85 ·10 -12 C 2 /N m 2.

Statement of the law:

the force of interaction between two point stationary charged bodies in a vacuum is directly proportional to the product of the charge modules and inversely proportional to the square of the distance between them.

This force is called Coulomb.

Coulomb's law in this formulation is valid only for point charged bodies, because only for them the concept of distance between charges has a certain meaning. There are no point charged bodies in nature. But if the distance between the bodies is many times greater than their size, then neither the shape nor the size of the charged bodies significantly, as experience shows, affects the interaction between them. In this case, the bodies can be considered as point bodies.

It is easy to find that two charged balls suspended on threads either attract each other or repel each other. It follows that the forces of interaction between two stationary point charged bodies are directed along the straight line connecting these bodies. Such forces are called central. If we denote by \(~\vec F_(1,2)\) the force acting on the first charge from the second, and by \(~\vec F_(2,1)\) the force acting on the second charge from the first (Fig. 1), then, according to Newton’s third law, \(~\vec F_(1,2) = -\vec F_(2,1)\) . Let us denote by \(\vec r_(1,2)\) the radius vector drawn from the second charge to the first (Fig. 2), then

\(~\vec F_(1,2) = k \cdot \dfrac(q_1 \cdot q_2)(r^3_(1,2)) \cdot \vec r_(1,2)\) . (2)

If the signs of the charges q 1 and q 2 are the same, then the direction of the force \(~\vec F_(1,2)\) coincides with the direction of the vector \(~\vec r_(1,2)\) ; otherwise, the vectors \(~\vec F_(1,2)\) and \(~\vec r_(1,2)\) are directed in opposite directions.

Knowing the law of interaction of point charged bodies, one can calculate the force of interaction of any charged bodies. To do this, bodies must be mentally broken down into such small elements that each of them can be considered a point. By adding geometrically the forces of interaction of all these elements with each other, we can calculate the resulting interaction force.

The discovery of Coulomb's law is the first concrete step in studying the properties of electric charge. The presence of an electric charge in bodies or elementary particles means that they interact with each other according to Coulomb's law. No deviations from the strict implementation of Coulomb's law have currently been detected.

Coulomb's experiment

The need to conduct Coulomb's experiments was caused by the fact that in the middle of the 18th century. A lot of high-quality data on electrical phenomena has accumulated. There was a need to give them a quantitative interpretation. Since the electrical interaction forces were relatively small, a serious problem arose in creating a method that would make it possible to make measurements and obtain the necessary quantitative material.

The French engineer and scientist C. Coulomb proposed a method for measuring small forces, which was based on the following experimental fact discovered by the scientist himself: the force arising during elastic deformation of a metal wire is directly proportional to the angle of twist, the fourth power of the diameter of the wire and inversely proportional to its length:

\(~F_(ynp) = k \cdot \dfrac(d^4)(l) \cdot \varphi\) ,

Where d– diameter, l– wire length, φ – twist angle. In the given mathematical expression, the proportionality coefficient k was determined empirically and depended on the nature of the material from which the wire was made.

This pattern was used in the so-called torsion balances. The created scales made it possible to measure negligible forces of the order of 5·10 -8 N.

Rice. 3

Torsion scales (Fig. 3, a) consisted of a light glass rocker 9 10.83 cm long, suspended on a silver wire 5 about 75 cm long, 0.22 cm in diameter. At one end of the rocker there was a gilded elderberry ball 8 , and on the other - a counterweight 6 - a paper circle dipped in turpentine. The upper end of the wire was attached to the head of the device 1 . There was also a sign here 2 , with the help of which the angle of twist of the thread was measured on a circular scale 3 . The scale was graduated. This entire system was housed in glass cylinders 4 And 11 . In the upper cover of the lower cylinder there was a hole into which a glass rod with a ball was inserted 7 at the end. In the experiments, balls with diameters ranging from 0.45 to 0.68 cm were used.

Before the start of the experiment, the head indicator was set to zero. Then the ball 7 charged from a pre-electrified ball 12 . When the ball touches 7 with movable ball 8 charge redistribution occurred. However, due to the fact that the diameters of the balls were the same, the charges on the balls were also the same 7 And 8 .

Due to the electrostatic repulsion of the balls (Fig. 3, b), the rocker 9 turned by some angle γ (on a scale 10 ). Using the head 1 this rocker returned to its original position. On a scale 3 pointer 2 allowed to determine the angle α twisting the thread. Total twist angle φ = γ + α . The force of interaction between the balls was proportional φ , i.e., by the angle of twist one can judge the magnitude of this force.

With a constant distance between the balls (it was recorded on a scale 10 in degree measure) the dependence of the force of electrical interaction of point bodies on the amount of charge on them was studied.

To determine the dependence of the force on the charge of the balls, Coulomb found a simple and ingenious way to change the charge of one of the balls. To do this, he connected a charged ball (balls 7 or 8 ) with the same size uncharged (ball 12 on the insulating handle). In this case, the charge was distributed equally between the balls, which reduced the charge under study by 2, 4, etc. times. The new value of the force at the new value of the charge was again determined experimentally. At the same time, it turned out that the force is directly proportional to the product of the charges of the balls:

\(~F \sim q_1 \cdot q_2\) .

The dependence of the strength of electrical interaction on distance was discovered as follows. After imparting a charge to the balls (they had the same charge), the rocker deviated at a certain angle γ . Then turn the head 1 this angle decreased to γ 1 . Total twist angle φ 1 = α 1 + (γ - γ 1)(α 1 – head rotation angle). When the angular distance of the balls is reduced to γ 2 total twist angle φ 2 = α 2 + (γ - γ 2) . It was noticed that if γ 1 = 2γ 2, TO φ 2 = 4φ 1, i.e., when the distance decreases by a factor of 2, the interaction force increases by a factor of 4. The moment of force increased by the same amount, since during torsional deformation the moment of force is directly proportional to the angle of twist, and therefore the force (the arm of the force remained unchanged). This leads to the following conclusion: The force of interaction between two charged balls is inversely proportional to the square of the distance between them:

\(~F \sim \dfrac(1)(r^2)\) .

Literature

  1. Myakishev G.Ya. Physics: Electrodynamics. 10-11 grades: textbook. for in-depth study of physics / G.Ya. Myakishev, A.Z. Sinyakov, B.A. Slobodskov. – M.: Bustard, 2005. – 476 p.
  2. Volshtein S.L. et al. Methods of physical science at school: A manual for teachers / S.L. Volshtein, S.V. Pozoisky, V.V. Usanov; Ed. S.L. Wolshtein. – Mn.: Nar. Asveta, 1988. – 144 p.

As a result of long observations, scientists have found that oppositely charged bodies attract, and similarly charged bodies, on the contrary, repel. This means that interaction forces arise between bodies. The French physicist C. Coulomb experimentally studied the patterns of interaction between metal balls and found that the force of interaction between two point electric charges will be directly proportional to the product of these charges and inversely proportional to the square of the distance between them:

Where k is a coefficient of proportionality, depending on the choice of units of measurement of physical quantities that are included in the formula, as well as on the environment in which the electric charges q 1 and q 2 are located. r is the distance between them.

From here we can conclude that Coulomb’s law will only be valid for point charges, that is, for such bodies whose sizes can be completely neglected in comparison with the distances between them.

In vector form, Coulomb's law will look like:

Where q 1 and q 2 are charges, and r is the radius vector connecting them; r = |r|.

The forces that act on the charges are called central. They are directed in a straight line connecting these charges, and the force acting from charge q 2 on charge q 1 is equal to the force acting from charge q 1 on charge q 2 and is opposite in sign.

To measure electrical quantities, two number systems can be used - the SI (basic) system and sometimes the CGS system can be used.

In the SI system, one of the main electrical quantities is the unit of current - ampere (A), then the unit of electric charge will be its derivative (expressed in terms of the unit of current). The SI unit of charge is the coulomb. 1 coulomb (C) is the amount of “electricity” passing through the cross-section of a conductor in 1 s at a current of 1 A, that is, 1 C = 1 A s.

Coefficient k in formula 1a) in SI is taken equal to:

And Coulomb’s law can be written in the so-called “rationalized” form:

Many equations describing magnetic and electrical phenomena contain a factor of 4π. However, if this factor is introduced into the denominator of Coulomb’s law, then it will disappear from most formulas of magnetism and electricity, which are very often used in practical calculations. This form of writing an equation is called rationalized.

The value ε 0 in this formula is the electrical constant.

The basic units of the GHS system are the GHS mechanical units (gram, second, centimeter). New basic units in addition to the above three are not introduced in the GHS system. The coefficient k in formula (1) is assumed to be equal to unity and dimensionless. Accordingly, Coulomb’s law in a non-rationalized form will look like:

In the CGS system, force is measured in dynes: 1 dyne = 1 g cm/s 2, and distance in centimeters. Let us assume that q = q 1 = q 2, then from formula (4) we obtain:

If r = 1 cm, and F = 1 dyne, then from this formula it follows that in the CGS system a unit of charge is taken to be a point charge, which (in a vacuum) acts on an equal charge, distant from it at a distance of 1 cm, with a force of 1 din. Such a unit of charge is called the absolute electrostatic unit of quantity of electricity (charge) and is denoted by CGS q. Its dimensions:

To calculate the value of ε 0, we compare the expressions for Coulomb’s law written in the SI and GHS systems. Two point charges of 1 C each, which are located at a distance of 1 m from each other, will interact with a force (according to formula 3):

In the GHS this force will be equal to:

The strength of interaction between two charged particles depends on the environment in which they are located. To characterize the electrical properties of various media, the concept of relative dielectric penetration ε was introduced.

The value of ε is a different value for different substances - for ferroelectrics its value lies in the range of 200 - 100,000, for crystalline substances from 4 to 3000, for glass from 3 to 20, for polar liquids from 3 to 81, for non-polar liquids from 1, 8 to 2.3; for gases from 1.0002 to 1.006.

The dielectric constant (relative) also depends on the ambient temperature.

If we take into account the dielectric constant of the medium in which the charges are placed, in SI Coulomb’s law takes the form:

Dielectric constant ε is a dimensionless quantity and it does not depend on the choice of units of measurement and for vacuum is considered equal to ε = 1. Then for vacuum Coulomb’s law takes the form:

Dividing expression (6) by (5) we get:

Accordingly, the relative dielectric constant ε shows how many times the interaction force between point charges in some medium, which are located at a distance r relative to each other, is less than in vacuum, at the same distance.

For the division of electricity and magnetism, the GHS system is sometimes called the Gaussian system. Before the advent of the SGS system, the SGSE (SGS electrical) systems operated for measuring electrical quantities and the SGSM (SGS magnetic) systems for measuring magnetic quantities. The first equal unit was taken to be the electrical constant ε 0, and the second equal to the magnetic constant μ 0.

In the SGS system, the formulas of electrostatics coincide with the corresponding formulas of the SGSE, and the formulas of magnetism, provided that they contain only magnetic quantities, coincide with the corresponding formulas in the SGSM.

But if the equation simultaneously contains both magnetic and electrical quantities, then this equation written in the Gaussian system will differ from the same equation, but written in the SGSM or SGSE system by the factor 1/s or 1/s 2 . The quantity c is equal to the speed of light (c = 3·10 10 cm/s) is called the electrodynamic constant.

Coulomb's law in the GHS system will have the form:

Example

Two absolutely identical drops of oil are missing one electron. The force of Newtonian attraction is balanced by the force of Coulomb repulsion. It is necessary to determine the radii of droplets if the distances between them significantly exceed their linear dimensions.

Solution

Since the distance r between the drops is significantly greater than their linear dimensions, the drops can be taken as point charges, and then the Coulomb repulsion force will be equal to:

Where e is the positive charge of the oil drop, equal to the charge of the electron.

The force of Newtonian attraction can be expressed by the formula:

Where m is the mass of the drop, and γ is the gravitational constant. According to the conditions of the problem, F k = F n, therefore:

The mass of a drop is expressed through the product of density ρ and volume V, that is, m = ρV, and the volume of a drop of radius R is equal to V = (4/3)πR 3, from which we obtain:

In this formula, the constants π, ε 0, γ are known; ε = 1; the electron charge e = 1.6·10 -19 C and the oil density ρ = 780 kg/m 3 (reference data) are also known. Substituting the numerical values ​​into the formula we get the result: R = 0.363·10 -7 m.

Just as in Newtonian mechanics gravitational interaction always takes place between bodies with masses, similarly in electrodynamics electrical interaction is characteristic of bodies with electric charges. Electric charge is indicated by the symbol “q” or “Q”.

One can even say that the concept of electric charge q in electrodynamics is somewhat similar to the concept of gravitational mass m in mechanics. But unlike gravitational mass, electric charge characterizes the property of bodies and particles to enter into force electromagnetic interactions, and these interactions, as you understand, are not gravitational.

Electric charges

Human experience in studying electrical phenomena contains many experimental results, and all these facts allowed physicists to come to the following clear conclusions regarding electric charges:

1. Electric charges are of two types - they can be conditionally divided into positive and negative.

2. Electrical charges can be transferred from one charged object to another: for example, by contacting bodies with each other - the charge between them can be divided. Moreover, the electric charge is not at all an obligatory component of the body: under different conditions, the same object may have a charge of different magnitude and sign, or the charge may be absent. Thus, the charge is not something inherent in the carrier, and at the same time, the charge cannot exist without the charge carrier.

3. While gravitating bodies are always attracted to each other, electric charges can both attract and repel each other. Like charges attract each other, like charges repel each other.

The law of conservation of electric charge is a fundamental law of nature, it sounds like this: “the algebraic sum of the charges of all bodies inside an isolated system remains constant.” This means that inside a closed system it is impossible for charges of only one sign to appear or disappear.

Today, the scientific point of view is that initially charge carriers are elementary particles. The elementary particles neutrons (electrically neutral), protons (positively charged) and electrons (negatively charged) form atoms.

Protons and neutrons make up the nuclei of atoms, and electrons form the shells of atoms. The moduli of the charges of the electron and proton are equal in magnitude to the elementary charge e, but the charges of these particles are opposite in sign.

As for the direct interaction of electric charges with each other, in 1785 the French physicist Charles Coulomb experimentally established and described this basic law of electrostatics, a fundamental law of nature that does not follow from any other laws. The scientist in his work studied the interaction of stationary point charged bodies and measured the forces of their mutual repulsion and attraction.

Coulomb experimentally established the following: “The forces of interaction between stationary charges are directly proportional to the product of the modules and inversely proportional to the square of the distance between them.”

This is the formulation of Coulomb's Law. And although point charges do not exist in nature, only in relation to point charges can we talk about the distance between them, within the framework of this formulation of Coulomb’s Law.

In fact, if the distances between the bodies greatly exceed their sizes, then neither the size nor the shape of the charged bodies will particularly affect their interaction, which means that the bodies for this task can rightly be considered point-like.

Let's consider this example. Let's hang a couple of charged balls on strings. Since they are somehow charged, they will either repel each other or attract each other. Since the forces are directed along the straight line connecting these bodies, these forces are central.

To denote the forces acting on the part of each charge on the other, we write: F12 is the force of action of the second charge on the first, F21 is the force of action of the first charge on the second, r12 is the radius vector from the second point charge to the first. If the charges have the same sign, then the force F12 will be codirectional to the radius vector, but if the charges have different signs, F12 will be directed opposite to the radius vector.

Using the law of interaction of point charges (Coulomb's Law), you can now find the interaction force for any point charges or point charged bodies. If the bodies are not point-like, then they are mentally broken down into chalk elements, each of which could be mistaken for a point charge.

After finding the forces acting between all the small elements, these forces are added geometrically and the resulting force is found. Elementary particles also interact with each other according to Coulomb's Law, and to this day no violations of this fundamental law of electrostatics have been observed.

In modern electrical engineering there is no area where Coulomb’s Law does not work in one form or another. Starting with electric current, ending with a simply charged capacitor. Especially those areas that relate to electrostatics - they are 100% related to Coulomb's Law. Let's look at just a few examples.

The simplest case is the introduction of a dielectric. The force of interaction of charges in a vacuum is always greater than the force of interaction of the same charges under conditions when some kind of dielectric is located between them.

The dielectric constant of a medium is precisely the quantity that allows us to quantify the values ​​of forces, regardless of the distance between the charges and their magnitudes. It is enough to divide the force of interaction of charges in a vacuum by the dielectric constant of the introduced dielectric - we obtain the force of interaction in the presence of the dielectric.

Complex research equipment - charged particle accelerator. The operation of charged particle accelerators is based on the phenomenon of interaction between the electric field and charged particles. The electric field does work in the accelerator, increasing the energy of the particle.

If we consider here the accelerated particle as a point charge, and the action of the accelerating electric field of the accelerator as the total force from other point charges, then in this case Coulomb’s Law is fully observed. The magnetic field only directs the particle by the Lorentz force, but does not change its energy, it only sets the trajectory for the movement of particles in the accelerator.

Protective electrical structures. Important electrical installations are always equipped with such a simple thing at first glance as a lightning rod. And a lightning rod cannot do its work without observing Coulomb’s Law. During a thunderstorm, large induced charges appear on Earth - according to Coulomb's Law, they are attracted in the direction of the thundercloud. This results in a strong electric field on the Earth's surface.

The intensity of this field is especially high near sharp conductors, and therefore a corona discharge is ignited at the pointed end of the lightning rod - a charge from the Earth tends, in obedience to Coulomb's Law, to be attracted to the opposite charge of a thundercloud.

The air near the lightning rod is highly ionized as a result of a corona discharge. As a result, the electric field strength near the tip decreases (as well as inside any conductor), induced charges cannot accumulate on the building and the likelihood of lightning occurring is reduced. If lightning happens to strike the lightning rod, the charge will simply go into the Earth and will not damage the installation.

Publications based on materials by D. Giancoli. "Physics in two volumes" 1984 Volume 2.

There is a force between electric charges. How does it depend on the magnitude of the charges and other factors?
This question was explored in the 1780s by the French physicist Charles Coulomb (1736-1806). He used torsion balances very similar to those used by Cavendish to determine the gravitational constant.
If a charge is applied to a ball at the end of a rod suspended on a thread, the rod is slightly deflected, the thread twists, and the angle of rotation of the thread will be proportional to the force acting between the charges (torsion balance). Using this device, Coulomb determined the dependence of force on the size of charges and the distance between them.

At that time, there were no instruments to accurately determine the amount of charge, but Coulomb was able to prepare small balls with a known charge ratio. If a charged conducting ball, he reasoned, is brought into contact with exactly the same uncharged ball, then the charge present on the first ball, due to symmetry, will be distributed equally between the two balls.
This gave him the ability to receive charges of 1/2, 1/4, etc. from the original one.
Despite some difficulties associated with the induction of charges, Coulomb was able to prove that the force with which one charged body acts on another small charged body is directly proportional to the electric charge of each of them.
In other words, if the charge of any of these bodies is doubled, the force will also be doubled; if the charges of both bodies are doubled at the same time, the force will become four times greater. This is true provided that the distance between the bodies remains constant.
By changing the distance between bodies, Coulomb discovered that the force acting between them is inversely proportional to the square of the distance: if the distance, say, doubles, the force becomes four times less.

So, Coulomb concluded, the force with which one small charged body (ideally a point charge, i.e. a body like a material point that has no spatial dimensions) acts on another charged body is proportional to the product of their charges Q 1 and Q 2 and is inversely proportional to the square of the distance between them:

Here k- proportionality coefficient.
This relationship is known as Coulomb's law; its validity has been confirmed by careful experiments, much more accurate than Coulomb's original, difficult to reproduce experiments. The exponent 2 is currently established with an accuracy of 10 -16, i.e. it is equal to 2 ± 2×10 -16.

Since we are now dealing with a new quantity - electric charge, we can select a unit of measurement so that the constant k in the formula is equal to one. Indeed, such a system of units was widely used in physics until recently.

We are talking about the CGS system (centimeter-gram-second), which uses the electrostatic charge unit SGSE. By definition, two small bodies, each with a charge of 1 SGSE, located at a distance of 1 cm from each other, interact with a force of 1 dyne.

Now, however, charge is most often expressed in the SI system, where its unit is the coulomb (C).
We will give the exact definition of a coulomb in terms of electric current and magnetic field later.
In the SI system the constant k has the magnitude k= 8.988×10 9 Nm 2 / Cl 2.

The charges arising during electrification by friction of ordinary objects (combs, plastic rulers, etc.) are in the order of magnitude a microcoulomb or less (1 µC = 10 -6 C).
The electron charge (negative) is approximately 1.602×10 -19 C. This is the smallest known charge; it has a fundamental meaning and is represented by the symbol e, it is often called the elementary charge.
e= (1.6021892 ± 0.0000046)×10 -19 C, or e≈ 1.602×10 -19 Cl.

Since a body cannot gain or lose a fraction of an electron, the total charge of the body must be an integer multiple of the elementary charge. They say that the charge is quantized (that is, it can take only discrete values). However, since the electron charge e is very small, we usually do not notice the discreteness of macroscopic charges (a charge of 1 µC corresponds to approximately 10 13 electrons) and consider the charge to be continuous.

The Coulomb formula characterizes the force with which one charge acts on another. This force is directed along the line connecting the charges. If the signs of the charges are the same, then the forces acting on the charges are directed in opposite directions. If the signs of the charges are different, then the forces acting on the charges are directed towards each other.
Note that, in accordance with Newton's third law, the force with which one charge acts on another is equal in magnitude and opposite in direction to the force with which the second charge acts on the first.
Coulomb's law can be written in vector form, similar to Newton's law of universal gravitation:

Where F 12 - vector of force acting on the charge Q 1 charge side Q 2,
- distance between charges,
- unit vector directed from Q 2 k Q 1.
It should be borne in mind that the formula is applicable only to bodies the distance between which is significantly greater than their own dimensions. Ideally, these are point charges. For bodies of finite size, it is not always clear how to calculate the distance r between them, especially since the charge distribution may be non-uniform. If both bodies are spheres with a uniform charge distribution, then r means the distance between the centers of the spheres. It is also important to understand that the formula determines the force acting on a given charge from a single charge. If the system includes several (or many) charged bodies, then the resulting force acting on a given charge will be the resultant (vector sum) of the forces acting on the part of the remaining charges. The constant k in the Coulomb Law formula is usually expressed in terms of another constant, ε 0 , the so-called electrical constant, which is related to k ratio k = 1/(4πε 0). Taking this into account, Coulomb's law can be rewritten as follows:

where with the highest accuracy today

or rounded

Writing most other equations of electromagnetic theory is simplified by using ε 0 , because the 4π the final result is often shortened. Therefore, we will generally use Coulomb's Law, assuming that:

Coulomb's law describes the force acting between two charges at rest. When charges move, additional forces are created between them, which we will discuss in subsequent chapters. Here only charges at rest are considered; This section of the study of electricity is called electrostatics.

To be continued. Briefly about the following publication:

Electric field is one of two components of the electromagnetic field, which is a vector field that exists around bodies or particles with an electric charge, or that arises when the magnetic field changes.

Comments and suggestions are accepted and welcome!

Coulomb's Law is a law that describes the interaction forces between point electric charges.

It was discovered by Charles Coulomb in 1785. After conducting a large number of experiments with metal balls, Charles Coulomb gave the following formulation of the law:

The modulus of the force of interaction between two point charges in a vacuum is directly proportional to the product of the moduli of these charges and inversely proportional to the square of the distance between them

Otherwise: Two point charges in a vacuum act on each other with forces that are proportional to the product of the moduli of these charges, inversely proportional to the square of the distance between them and directed along the straight line connecting these charges. These forces are called electrostatic (Coulomb).

It is important to note that in order for the law to be true, it is necessary:

  1. point-like charges - that is, the distance between charged bodies is much larger than their sizes - however, it can be proven that the force of interaction of two volumetrically distributed charges with spherically symmetrical non-intersecting spatial distributions is equal to the force of interaction of two equivalent point charges located at centers of spherical symmetry;
  2. their immobility. Otherwise, additional effects come into force: the magnetic field of a moving charge and the corresponding additional Lorentz force acting on another moving charge;
  3. interaction in a vacuum.

However, with some adjustments, the law is also valid for interactions of charges in a medium and for moving charges.

In vector form in the formulation of C. Coulomb, the law is written as follows:

where is the force with which charge 1 acts on charge 2; - magnitude of charges; — radius vector (vector directed from charge 1 to charge 2, and equal, in absolute value, to the distance between charges — ); — proportionality coefficient. Thus, the law indicates that like charges repel (and unlike charges attract).

Coefficient k

In the SGSE, the unit of measurement of charge is chosen in such a way that the coefficient k equal to one.

In the International System of Units (SI), one of the basic units is the unit of electric current, the ampere, and the unit of charge, the coulomb, is a derivative of it. The ampere value is defined in such a way that k= c2·10-7 H/m = 8.9875517873681764·109 N·m2/Cl2 (or Ф−1·m). SI coefficient k is written as:

where ≈ 8.854187817·10−12 F/m is the electrical constant.

In a homogeneous isotropic substance, the relative dielectric constant of the medium ε is added to the denominator of the formula.

Coulomb's law in quantum mechanics

In quantum mechanics, Coulomb's law is formulated not using the concept of force, as in classical mechanics, but using the concept of potential energy of the Coulomb interaction. In the case when the system considered in quantum mechanics contains electrically charged particles, terms are added to the Hamiltonian operator of the system, expressing the potential energy of the Coulomb interaction, as it is calculated in classical mechanics.

Thus, the Hamilton operator of an atom with a nuclear charge Z has the form:

j)\frac(e^2)(r_(ij))" src="http://upload.wikimedia.org/math/d/0/8/d081b99fac096b0e0c5b4290a9573794.png">.

Here m- electron mass, e is its charge, is the absolute value of the radius vector j th electron, . The first term expresses the kinetic energy of electrons, the second term expresses the potential energy of the Coulomb interaction of electrons with the nucleus, and the third term expresses the potential Coulomb energy of mutual repulsion of electrons. The summation in the first and second terms is carried out over all N electrons. In the third term, the summation occurs over all pairs of electrons, with each pair occurring once.

Coulomb's law from the point of view of quantum electrodynamics

According to quantum electrodynamics, the electromagnetic interaction of charged particles occurs through the exchange of virtual photons between particles. The uncertainty principle for time and energy allows for the existence of virtual photons for the time between the moments of their emission and absorption. The smaller the distance between charged particles, the less time it takes virtual photons to overcome this distance and, therefore, the greater the energy of virtual photons allowed by the uncertainty principle. At small distances between charges, the uncertainty principle allows the exchange of both long- and short-wave photons, and at large distances only long-wave photons participate in the exchange. Thus, using quantum electrodynamics, Coulomb's law can be derived.

Story

For the first time, G.V. Richman proposed to study experimentally the law of interaction of electrically charged bodies in 1752-1753. He intended to use the “pointer” electrometer he had designed for this purpose. The implementation of this plan was prevented by the tragic death of Richman.

In 1759, F. Epinus, a professor of physics at the St. Petersburg Academy of Sciences, who took over Richmann's chair after his death, first suggested that charges should interact in inverse proportion to the square of the distance. In 1760, a brief message appeared that D. Bernoulli in Basel had established the quadratic law using an electrometer he had designed. In 1767, Priestley noted in his History of Electricity that Franklin's discovery of the absence of an electric field inside a charged metal ball might mean that "electrical attraction follows exactly the same law as gravity, that is, the square of the distance". The Scottish physicist John Robison claimed (1822) to have discovered in 1769 that balls of equal electrical charge repel with a force inversely proportional to the square of the distance between them, and thus anticipated the discovery of Coulomb's law (1785).

About 11 years before Coulomb, in 1771, the law of interaction of charges was experimentally discovered by G. Cavendish, but the result was not published and remained unknown for a long time (over 100 years). Cavendish's manuscripts were presented to D. C. Maxwell only in 1874 by one of Cavendish's descendants at the inauguration of the Cavendish Laboratory and published in 1879.

Coulomb himself studied the torsion of threads and invented the torsion balance. He discovered his law by using them to measure the interaction forces of charged balls.

Coulomb's law, superposition principle and Maxwell's equations

Coulomb's law and the principle of superposition for electric fields are completely equivalent to Maxwell's equations for electrostatics and. That is, Coulomb's law and the superposition principle for electric fields are satisfied if and only if Maxwell's equations for electrostatics are satisfied and, conversely, Maxwell's equations for electrostatics are satisfied if and only if Coulomb's law and the superposition principle for electric fields are satisfied.

Degree of accuracy of Coulomb's law

Coulomb's law is an experimentally established fact. Its validity has been repeatedly confirmed by increasingly accurate experiments. One direction of such experiments is to test whether the exponent differs r in the law from 2. To find this difference, we use the fact that if the power is exactly equal to two, then there is no field inside the cavity in the conductor, whatever the shape of the cavity or conductor.

Experiments carried out in 1971 in the USA by E. R. Williams, D. E. Voller and G. A. Hill showed that the exponent in Coulomb's law is equal to 2 to within .

To test the accuracy of Coulomb's law at intra-atomic distances, W. Yu. Lamb and R. Rutherford in 1947 used measurements of the relative positions of hydrogen energy levels. It was found that even at distances of the order of atomic 10−8 cm, the exponent in Coulomb's law differs from 2 by no more than 10−9.

The coefficient in Coulomb's law remains constant with an accuracy of 15·10−6.

Amendments to Coulomb's law in quantum electrodynamics

At short distances (on the order of the Compton electron wavelength, ≈3.86·10−13 m, where is the electron mass, is Planck’s constant, and is the speed of light), the nonlinear effects of quantum electrodynamics become significant: the exchange of virtual photons is superimposed on the generation of virtual electron-positron (and also muon-antimuon and taon-antitaon) pairs, and the influence of screening is reduced (see renormalization). Both effects lead to the appearance of exponentially decreasing order terms in the expression for the potential energy of interaction of charges and, as a result, to an increase in the interaction force compared to that calculated by Coulomb’s law. For example, the expression for the potential of a point charge in the SGS system, taking into account first-order radiation corrections, takes the form:

where is the Compton wavelength of the electron, is the fine structure constant and . At distances of the order of ~ 10−18 m, where is the mass of the W boson, electroweak effects come into play.

In strong external electromagnetic fields, constituting a noticeable fraction of the vacuum breakdown field (of the order of ~1018 V/m or ~109 Tesla, such fields are observed, for example, near some types of neutron stars, namely magnetars), Coulomb’s law is also violated due to Delbrück scattering of exchange photons on external field photons and other, more complex nonlinear effects. This phenomenon reduces the Coulomb force not only on a micro but also on a macro scale; in particular, in a strong magnetic field, the Coulomb potential does not fall in inverse proportion to distance, but exponentially.

Coulomb's law and vacuum polarization

The phenomenon of vacuum polarization in quantum electrodynamics consists in the formation of virtual electron-positron pairs. A cloud of electron-positron pairs screens the electrical charge of the electron. Screening increases with increasing distance from the electron; as a result, the effective electric charge of the electron is a decreasing function of distance. The effective potential created by an electron with an electric charge can be described by a dependence of the form . The effective charge depends on the distance according to the logarithmic law:

- so-called fine structure constant ≈7.3·10−3;

- so-called classical electron radius ≈2.8·10−13 cm.

Juhling effect

The phenomenon of deviation of the electrostatic potential of point charges in a vacuum from the value of Coulomb's law is known as the Juhling effect, which was the first to calculate deviations from Coulomb's law for the hydrogen atom. The Uehling effect provides a correction to the Lamb shift of 27 MHz.

Coulomb's law and superheavy nuclei

In a strong electromagnetic field near superheavy nuclei with a charge of 170" src="http://upload.wikimedia.org/math/0/d/7/0d7b5476a5437d2a99326cf04b131458.png"> a restructuring of the vacuum occurs, similar to a conventional phase transition. This leads to corrections to Coulomb's law.

The significance of Coulomb's law in the history of science

Coulomb's law is the first open quantitative law for electromagnetic phenomena formulated in mathematical language. The modern science of electromagnetism began with the discovery of Coulomb's law.