Like the concept of gravitational mass of a body in Newtonian mechanics, the concept of charge in electrodynamics is the primary, basic concept.
Electric charge is a physical quantity that characterizes the property of particles or bodies to enter into electromagnetic force interactions.
Electric charge is usually represented by the letters q or Q.
The totality of all known experimental facts allows us to draw the following conclusions:
There are two types of electric charges, conventionally called positive and negative.
Charges can be transferred (for example, by direct contact) from one body to another. Unlike body mass, electric charge is not an integral characteristic of a given body. The same body under different conditions can have a different charge.
Like charges repel, unlike charges attract. This also reveals the fundamental difference between electromagnetic forces and gravitational ones. Gravitational forces are always attractive forces.
One of the fundamental laws of nature is the experimentally established law of conservation of electric charge .
In an isolated system, the algebraic sum of the charges of all bodies remains constant:
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q 1 + q 2 + q 3 + ... +qn= const. |
The law of conservation of electric charge states that in a closed system of bodies processes of creation or disappearance of charges of only one sign cannot be observed.
From a modern point of view, charge carriers are elementary particles. All ordinary bodies consist of atoms, which include positively charged protons, negatively charged electrons and neutral particles - neutrons. Protons and neutrons are part of atomic nuclei, electrons form the electron shell of atoms. The electric charges of a proton and an electron are exactly the same in magnitude and equal to the elementary charge e.
In a neutral atom, the number of protons in the nucleus is equal to the number of electrons in the shell. This number is called atomic number . An atom of a given substance may lose one or more electrons or gain an extra electron. In these cases, the neutral atom turns into a positively or negatively charged ion.
Charge can be transferred from one body to another only in portions containing an integer number of elementary charges. Thus, the electric charge of a body is a discrete quantity:
Physical quantities that can only take a discrete series of values are called quantized . Elementary charge e is a quantum (smallest portion) of electric charge. It should be noted that in modern physics of elementary particles the existence of so-called quarks is assumed - particles with a fractional charge and However, quarks have not yet been observed in a free state.
In common laboratory experiments, a electrometer ( or electroscope) - a device consisting of a metal rod and a pointer that can rotate around a horizontal axis (Fig. 1.1.1). The arrow rod is isolated from the metal body. When a charged body comes into contact with the electrometer rod, electric charges of the same sign are distributed over the rod and the pointer. Electrical repulsion forces cause the needle to rotate through a certain angle, by which one can judge the charge transferred to the electrometer rod.
The electrometer is a rather crude instrument; it does not allow one to study the forces of interaction between charges. The law of interaction of stationary charges was first discovered by the French physicist Charles Coulomb in 1785. In his experiments, Coulomb measured the forces of attraction and repulsion of charged balls using a device he designed - a torsion balance (Fig. 1.1.2), which was distinguished by extremely high sensitivity. For example, the balance beam was rotated 1° under the influence of a force of the order of 10 -9 N.
The idea of the measurements was based on Coulomb's brilliant guess that if a charged ball is brought into contact with exactly the same uncharged one, then the charge of the first will be divided equally between them. Thus, a way was indicated to change the charge of the ball by two, three, etc. times. In Coulomb's experiments, the interaction between balls whose dimensions were much smaller than the distance between them was measured. Such charged bodies are usually called point charges.
Point charge called a charged body, the dimensions of which can be neglected in the conditions of this problem.
Based on numerous experiments, Coulomb established the following law:
The interaction forces between stationary charges are directly proportional to the product of the charge moduli and inversely proportional to the square of the distance between them:
Interaction forces obey Newton's third law:
They are repulsive forces with the same signs of charges and attractive forces with different signs (Fig. 1.1.3). The interaction of stationary electric charges is called electrostatic or Coulomb interaction. The branch of electrodynamics that studies the Coulomb interaction is called electrostatics .
Coulomb's law is valid for point charged bodies. In practice, Coulomb's law is well satisfied if the sizes of charged bodies are much smaller than the distance between them.
Proportionality factor k in Coulomb's law depends on the choice of system of units. In the International SI System, the unit of charge is taken to be pendant(Cl).
Pendant is a charge passing in 1 s through the cross section of a conductor at a current strength of 1 A. The unit of current (Ampere) in SI is, along with units of length, time and mass basic unit of measurement.
Coefficient k in the SI system it is usually written as:
Where - electrical constant .
In the SI system, the elementary charge e equal to:
Experience shows that the Coulomb interaction forces obey the superposition principle:
If a charged body interacts simultaneously with several charged bodies, then the resulting force acting on a given body is equal to the vector sum of the forces acting on this body from all other charged bodies.
Rice. 1.1.4 explains the principle of superposition using the example of the electrostatic interaction of three charged bodies.
The principle of superposition is a fundamental law of nature. However, its use requires some caution when we are talking about the interaction of charged bodies of finite sizes (for example, two conducting charged balls 1 and 2). If a third charged ball is brought to a system of two charged balls, then the interaction between 1 and 2 will change due to charge redistribution.
The principle of superposition states that when given (fixed) charge distribution on all bodies, the forces of electrostatic interaction between any two bodies do not depend on the presence of other charged bodies.
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's assume that q = q 1 = q 2 , then from formula (4) we get:
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.
Coulomb's Law is a law that describes the interaction forces between point electric charges.
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 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:
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;
their immobility. Otherwise, additional effects take effect: a magnetic field moving charge and the corresponding additional Lorentz force, acting on another moving charge;
interaction in 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).
IN SSSE unit charge is chosen in such a way that the coefficient k equal to one.
IN International System of Units (SI) one of the basic units is the unit electric current strength ampere, and the unit of charge is pendant- a derivative of it. The ampere value is defined in such a way that k= c 2 10 −7 Gn/m = 8.9875517873681764 10 9 N m 2 / Cl 2 (or Ф −1 m). SI coefficient k is written as:
where ≈ 8.854187817·10 −12 F/m - electrical constant.
The interaction of electric charges is described by Coulomb's law, which states that the force of interaction between two point charges at rest in a vacuum is equal to
where the quantity is called the electrical constant, the dimension of the quantity is reduced to the ratio of the dimension of length to the dimension of electrical capacitance (Farad). Electric charges are of two types, which are conventionally called positive and negative. As experience shows, charges attract if they are opposite and repel if they are like.
Any macroscopic body contains a huge amount of electrical charges, since they are part of all atoms: electrons are negatively charged, protons that are part of atomic nuclei are positively charged. However, most of the bodies we deal with are not charged, since the number of electrons and protons that make up the atoms is the same, and their charges are exactly the same in absolute value. However, bodies can be charged by creating an excess or deficiency of electrons in them compared to protons. To do this, you need to transfer the electrons that are part of a body to another body. Then the first will have a lack of electrons and, accordingly, a positive charge, while the second will have a negative charge. This kind of process occurs, in particular, when bodies rub against each other.
If the charges are in a certain medium that occupies the entire space, then the force of their interaction is weakened compared to the force of their interaction in a vacuum, and this weakening does not depend on the magnitude of the charges and the distance between them, but depends only on the properties of the medium. The characteristic of a medium, which shows how many times the force of interaction of charges in this medium is weakened compared to the force of their interaction in a vacuum, is called the dielectric constant of this medium and, as a rule, is denoted by the letter. The Coulomb formula in a medium with dielectric constant takes the form
If there are not two, but a larger number of point charges, to find the forces acting in this system, a law is used, which is called the principle superposition 1. The principle of superposition states that in order to find the force acting on one of the charges (for example, charge) in a system of three point charges, the following must be done. First, you need to mentally remove the charge and, according to Coulomb’s law, find the force acting on the charge from the remaining charge. Then you should remove the charge and find the force acting on the charge from the charge. The vector sum of the received forces will give the desired force.
The principle of superposition provides a recipe for searching for the force of interaction between non-point charged bodies. You should mentally break each body into parts that can be considered point parts, use Coulomb’s law to find the force of their interaction with the point parts into which the second body is broken, and sum up the resulting vectors. It is clear that such a procedure is mathematically very complex, if only because it is necessary to add an infinite number of vectors. Methods for such summation have been developed in mathematical analysis, but they are not included in the school physics course. Therefore, if such a problem is encountered, then the summation in it should be easily performed on the basis of certain symmetry considerations. For example, from the described summation procedure it follows that the force acting on a point charge placed at the center of a uniformly charged sphere is zero.
In addition, the student must know (without derivation) the formulas for the force acting on a point charge from a uniformly charged sphere and an infinite plane. If there is a sphere of radius , uniformly charged with charge , and a point charge located at a distance from the center of the sphere, then the magnitude of the interaction force is equal to
if the charge is inside (and not necessarily in the center). From formulas (17.4), (17.5) it follows that the sphere outside creates the same electric field as its entire charge placed in the center, and inside it creates zero.
If there is a very large plane with an area uniformly charged with a charge and a point charge, then the force of their interaction is equal to
where the quantity has the meaning of the surface charge density of the plane. As follows from formula (17.6), the force of interaction between a point charge and a plane does not depend on the distance between them. Let us draw the reader's attention to the fact that formula (17.6) is approximate and “works” the more accurately, the further the point charge is from its edges. Therefore, when using formula (17.6), it is often said that it is valid within the framework of neglecting “edge effects”, i.e. when the plane is considered infinite.
Let us now consider solving the data in the first part of the book of problems.
According to Coulomb’s law (17.1), the magnitude of the interaction force between two charges from tasks 17.1.1 expressed by the formula
Charges repel (answer) 2 ).
Since a drop of water from tasks 17.1.2 has a charge ( – the charge of a proton), then it has an excess of electrons compared to protons. This means that with the loss of three electrons, their excess will decrease, and the charge of the droplet will become equal (answer 2 ).
According to Coulomb’s law (17.1), the magnitude of the interaction force between two charges increases by a factor of the distance between them will decrease by a factor ( problem 17.1.3- answer 4 ).
If the charges of two point bodies are increased by a factor with a constant distance between them, then the force of their interaction, as follows from Coulomb’s law (17.1), will increase by a factor ( problem 17.1.4- answer 3 ).
When one charge increases by 2 times, and the second by 4, the numerator of Coulomb's law (17.1) increases by 8 times, and when the distance between charges increases by 8 times, the denominator increases by 64 times. Therefore, the force of interaction between charges from problems 17.1.5 will decrease by 8 times (answer 4 ).
When filling space with a dielectric medium with dielectric constant = 10, the force of interaction of charges according to Coulomb’s law in the medium (17.3) will decrease by 10 times ( problem 17.1.6- answer 2 ).
The Coulomb interaction force (17.1) acts on both the first and second charges, and since their masses are the same, the accelerations of the charges, as follows from Newton’s second law, are the same at any time ( problem 17.1.7- answer 3 ).
A similar problem, but the masses of the balls are different. Therefore, with the same force, the acceleration of a ball with a smaller mass is 2 times greater than the acceleration of a ball with a smaller mass, and this result does not depend on the magnitude of the charges of the balls ( problem 17.1.8- answer 2 ).
Since the electron is negatively charged, it will be repelled from the ball ( problem 17.1.9). But since the initial speed of the electron is directed towards the ball, it will move in that direction, but its speed will decrease. At some point it will stop for a moment, and then will move away from the ball with increasing speed (answer 4 ).
In a system of two charged balls connected by a thread ( problem 17.1.10), only internal forces act. Therefore, the system will be at rest and the equilibrium conditions of the balls can be used to find the tension force of the thread. Since each of them is affected only by the Coulomb force and the tension force of the thread, we conclude from the equilibrium condition that these forces are equal in magnitude.
This value will be equal to the tension force of the threads (answer 4 ). Note that considering the equilibrium condition of the central charge would not help to find the tension force, but would lead to the conclusion that the tension forces of the threads are the same (however, this conclusion is already obvious due to the symmetry of the problem).
To find the force acting on the charge - in problem 17.2.2, we use the principle of superposition. The charge is affected by attractive forces towards the left and right charges (see figure). Since the distances from the charge - to the charges are the same, the moduli of these forces are equal to each other and they are directed at the same angles to the straight line connecting the charge - with the middle of the segment -. Therefore, the force acting on the charge is directed vertically downward (the vector of the resulting force is highlighted in bold in the figure; answer 4 ).
(answer 3 ).
From formula (17.6) we conclude that the correct answer is in problem 17.2.5 - 4 . IN problem 17.2.6 you need to use the formula for the force of interaction between a point charge and a sphere (formulas (17.4), (17.5)). We have = 0 (answer 3 ).
IN problem 17.2.7 it is necessary to apply the principle of superposition to the two spheres. The principle of superposition states that the interaction of each pair of charges is independent of the presence of other charges. Therefore, each sphere acts on a point charge independently of the other sphere, and to find the resultant force it is necessary to add the forces from the first and second spheres. Since the point charge is located inside the outer sphere, it does not act on it (see formula (17.5)), the inner one acts with a force
Where . Therefore, the resulting force is equal to this expression (answer 2 )
IN problem 17.2.8 the principle of superposition should also be used. If a charge is placed at point , then the forces acting on it from charges and are directed to the left. Therefore, according to the superposition principle, we have for the resultant force
where are the distances from the charges to the points under study. If we place a positive charge at point , then the forces will be directed in the opposite direction, and based on the principle of superposition we find the resulting force
From these formulas it follows that the greatest force will be at the point - answer 1 .
Let, for definiteness, the charges of the balls and in problem 17.2.9 are positive. Since the balls are identical, the charges after their connection are distributed between them evenly and to compare forces, you need to compare the values with each other
which are the products of the charges of the balls before and after their connection. After extracting the square root, comparison (1) comes down to comparing the geometric mean and the arithmetic mean of two numbers. And since the arithmetic mean of any two numbers is greater than their geometric mean, the force of interaction between the balls will increase regardless of the magnitude of their charges (answer 1 ).
Problem 17.2.10 very similar to the previous one, but the answer is different. By direct verification it is easy to verify that the force can either increase or decrease depending on the magnitude of the charges. For example, if the charges are equal in magnitude, then after connecting the balls their charges will become zero, so the force of their interaction will also be zero, which will therefore decrease. If one of the initial charges is zero, then after the balls touch, the charge of one of them will be distributed equally between the balls, and the force of their interaction will increase. Thus, the correct answer in this problem is 3 .