What is the electric field strength? What is electric field strength. What is electrical capacitance

As you already know from the basic school physics course, the electrical interaction of charged bodies is carried out through electric field: Each charged body creates an electric field around itself, which acts on other charged bodies. The concept of the electric field was introduced by the English scientist Michael Faraday in the first half of the 19th century.

The electric field at a given point in space can be characterized by the force exerted by this field on a point charge placed in this point. (This charge must be small enough so that the field it creates does not change the distribution of charges that create the field.)

As experience shows, the force acting on charge q is proportional to the magnitude of this charge. Consequently, the ratio of force to charge does not depend on the magnitude of the charge and characterizes the electric field itself.

The electric field strength at a given point is called physical quantity, equal to the ratio of the force acting from the field on a charge q placed at a given point in the field to the magnitude of this charge:

Field strength – vector quantity. Its direction at each point coincides with the direction of the force acting on the positive charge placed at this point.

The unit of field strength is 1 N/C. 1 N/C – low tension. For example, the electric field strength near the Earth's surface, due to the Earth's electric charge, is approximately 130 N/C.

If the field strength at a given point is known, then the force acting on a charge q placed at this point can be found using the formula

From formulas (1) and (2) it follows that the direction of the field strength at a given point coincides with the direction of the force acting on a positive charge placed at this point.

Point charge field strength

If you introduce another positive charge into the field of a positive point charge Q, it will be repelled from the charge Q.

Consequently, the field strength of a positive point charge at all points in space is directed away from this charge. Figure 51.1 shows the field strength vectors of a point charge at some points. It can be seen that the modulus of the field strength decreases with distance from the charge.

1. Explain why the modulus of the field strength of a point charge Q at a distance r from the charge is expressed by the formula

Clue. Use Coulomb's law and the definition of field strength.

2. What is the field strength of a point charge of 2 nC at a distance of 2 m from it?

3. The modulus of the field strength of a point charge at a distance of 0.5 m from it is equal to 90 N/C. What could this charge be equal to?

Principle of field superposition

If a charge is in a field created by several charges, then each of these charges acts on a given charge independently of the others.

It follows that the resultant of the forces acting on a given charge from other charges is equal to the vector sum of the forces acting on a given charge from each of the other charges.

This means that the principle of field superposition is valid:

The field strength created by several charges is equal to the vector sum of the field strengths created by each of the charges:

Using the principle of superposition, you can find the field strength created by several charges.

4. Two point charges are located at a distance of 60 cm from each other. The modulus of each charge is 8 nC. What is the modulus of the field strength created by these charges:
a) at a point located in the middle of a segment connecting the charges, if the charges are of the same name? different names?
b) at a point located at a distance of 60 cm from each charge, if the charges are the same? different names?

For each of these cases, make a drawing in your notebook explaining the solution.

2. Lines of tension

Using the example of a point charge field (Fig. 51.1), one can notice that the electric field strength vectors in different points spaces are lined up along certain lines.

In the case of a point charge, these lines represent straight rays drawn from the point at which the charge is located. In a field created by several charges, these lines will be some curves, and the field strength at each point will be directed tangentially to one of these lines.

Imaginary lines, the tangents to which at each point coincide with the direction of the electric field strength, are called electric field strength lines.

Tension lines start at positive charges and end at negative charges. The density of tension lines is proportional to the tension modulus.

5. Explain why electric field lines cannot intersect.

Point charge fields

6. Explain why the electric field strength lines of positive and negative point charges have the form shown in Figures 51.2, a and 51.2, b.

7. Figure 51.3 shows the lines of field strength created by charges of equal magnitude (opposite and like). At some points, field strength vectors are shown for clarity.


a) Transfer the drawings to your notebook and mark the charge signs on them.
b) Draw in your notebook the lines of field strength created by two charges of the same name, which does not coincide with any of the given drawings.
c) What is the field strength at the central point of Figure 51.3, b (in the middle of the segment connecting the charges? Explain your answer using Coulomb’s law.

Field of a uniformly charged sphere

Figure 51.4 shows the electric field strength lines of a uniformly charged sphere.

We see that outside the sphere this field coincides with the field of a point charge, equal to the total charge of the sphere and located in the center of the sphere.
It can be proven that inside a charged sphere the field strength is zero. (Proof of this fact is beyond our scope.)

8. A sphere with a radius of 5 cm contains a charge of 6 nC. What is the field strength of this charge:
a) in the center of the sphere?
b) at a distance of 4 cm from the center of the sphere?
c) at a distance of 10 cm from the center of the sphere?
d) outside the sphere at a distance of 1 cm from the surface of the sphere closest to this point?

However, the electric field strength inside a charged sphere is not necessarily zero! If there is a charged body inside this sphere, then, according to the principle of superposition, the electric field strength is equal to the vector sum of the field strength created by the charge of this body and the field strength created by the charge of the sphere.

Inside a sphere, the field is created only by a charged body located inside the sphere, because the field strength created by a charged sphere inside the sphere is zero. And at any point outside the sphere, the field strength can be found by adding the vectors of the field strength created by a body located inside the sphere and the field created by the charge of the sphere.

9. There are two concentric (having a common center) spheres of radius 5 cm and 10 cm. The charge of the inner sphere is 6 nC, and the charge of the outer sphere is –9 nC. What is the magnitude of the field strength at a point located from the common center of the spheres at a distance equal to:
a) 3 cm; b) 6 cm; c) 8 cm; d) 12 cm; d) 20 cm?

Field of a uniformly charged plane

Figure 51.5 shows electric field lines near a uniformly charged flat plate.

We will assume that the dimensions of the plate are much larger than the distances from it to those points in space at which we consider the field strength. In such cases we speak of the field of a uniformly charged plane.

The field strength of a uniformly charged plane is practically the same (in magnitude and direction) at all points in space on one side of the plane. The intensity lines of this field are parallel straight lines, perpendicular planes and located at equal distances from each other. Such an electric field is called uniform.

On the other side of the plane, only the direction of the field strength changes, and its magnitude remains the same.

10. The electric field strength created by a large uniformly charged plate is 900 N/C. At a distance of 40 cm from the plate there is a point charge equal in modulus to 1 nC.
a) At what distance from a point charge is the modulus of its field strength equal to the modulus of the field strength of the plate?
b) At what distance from the plane is the resulting field strength of the plane and the point charge equal to zero if the sign of the point charge coincides with the sign of the plane charge? What if the sign of a point charge is opposite to the sign of the plane charge?

Field of two oppositely charged flat plates

Let's take two identical uniformly charged plates, the charges of which are equal in magnitude but opposite in sign. Let's place the plates parallel to each other at a small distance from each other (Fig. 51.6).

11. Explain why in the space between the plates the field strength is 2 times greater than the field strength created by each of the plates, and outside the plates it is practically zero.
Clue. Use the principle of superposition of electric fields.

How to see tension lines?

Let's put experience
Let us place in an electric field small oblong-shaped bodies consisting of a dielectric - crystals, particles of semolina, finely cut hair, etc. In the electric field they are rotated so that their longer side is directed along the field strength vector. As a result, these bodies line up along lines of tension, making their shape visible. Figure 51.7 shows the resulting “pictures” of electric fields created by a charged ball (Fig. 51.7, a) and two differently charged balls (Fig. 51.7, b).

Additional questions and tasks

12. A small charged ball with a mass of 0.2 g is suspended on a thread in a uniform electric field, the intensity of which is directed horizontally and is equal in magnitude to 50 kN/C.
a) Draw on the drawing the equilibrium position of the ball and the forces acting on it.
b) What is the charge of the ball if the thread is deflected from the vertical at an angle of 30º?

13. What should the field strength be for a drop of water with a radius of 0.01 mm to be in equilibrium in this field, having lost 10 3 electrons? How should the field strength be directed?

Tension The electric field is a vector quantity, which means it has a numerical magnitude and direction. The magnitude of the electric field strength has its own dimension, which depends on the method of its calculation.

The electric force of interaction of charges is described as a non-contact action, and in other words, long-range action takes place, that is, action at a distance. In order to describe such long-range action, it is convenient to introduce the concept of an electric field and, with its help, explain the action at a distance.

Let's take an electric charge, which we will denote by the symbol Q. This electric charge creates an electric field, that is, it is the source of force. Since in the universe there is always at least one positive and at least one negative charge, which act on each other at any, even infinitely distant, distance, then any charge is source of strength, which means it is appropriate to describe the electric field they create. In our case, the charge Q is source electric field and we will consider it precisely as a source of the field.

Electric field strength source charge can be measured using any other charge located somewhere in its vicinity. The charge that is used to measure the electric field strength is called test charge, as it is used to test field strength. A test charge has a certain amount of charge and is indicated by the symbol q.

When placed trial charge into an electric field source of strength(charge Q), trial the charge will experience the action of an electrical force - either attraction or repulsion. Force can be denoted as is usually accepted in physics by the symbol F. Then the magnitude of the electric field can be defined simply as the ratio of the force to the magnitude trial charge.

If the electric field strength is indicated by the symbol E, then the equation can be rewritten in symbolic form as

The standard metric units for measuring electric field strength arise from its definition. Thus, the electric field strength is defined as a force equal to 1 Newton(H) divided by 1 Pendant(Cl). Electric field strength is measured in Newton/Coulomb or otherwise N/Kl. In the SI system it is also measured in Voltmeter. To understand the essence of such a subject, the dimension in the metric system is much more important. N/C, because this dimension reflects the origin of such a characteristic as field strength. The Volt/Meter notation makes the concept of field potential (Volt) basic, which is useful in some areas, but not in all.

The above example involves two charges Q (source) And q trial. Both of these charges are a source of force, but which one should be used in the above formula? There is only one charge in the formula and that is trial charge q(not source).

Doesn't depend on quantity trial charge q. This may seem confusing at first glance, if you really think about it. The trouble is that not everyone has the useful habit of thinking and remains in the so-called blissful ignorance. If you don’t think, then you won’t have this kind of confusion. So how does the electric field strength not depend on q, If q present in the equation? Great question! But if you think about it a little, you can answer this question. Increase in quantity trial charge q- say, 2 times - the denominator of the equation will also increase 2 times. But in accordance with Coulomb's Law, increasing the charge will also proportionally increase the generated force F. The charge will increase 2 times, then the strength F will increase by the same amount. Since the denominator in the equation increases by a factor of two (or three, or four), the numerator will increase by the same amount. These two changes cancel each other out, so we can safely say that the electric field strength does not depend on the amount trial charge.

Thus, no matter how many trial charge q used in the equation, electric field strength E at any given point around the charge Q (source) will be the same when measured or calculated.

Learn more about the electric field strength formula

Above we touched on the definition of electric field strength in how it is measured. Now we will try to explore a more detailed equation with variables in order to more clearly imagine the very essence of calculating and measuring the electric field strength. From the equation we can see exactly what is affected and what is not. To do this, we first need to return to the equation of Coulomb's Law.

Coulomb's law states that electric force F between two charges is directly proportional to the product of the number of these charges and inversely proportional to the square of the distance between their centers.

If we add our two charges into the equation of Coulomb's Law Q (source) And q (trial charge), then we get the following entry:

If the expression for electric force F how is it determined Coulomb's law substitute into the equation for electric field strength E which is given above, then we get the following equation:

note that trial charge q was reduced, that is, removed from both the numerator and the denominator. New formula for electric field strength E expresses field strength in terms of two variables that influence it. Electric field strength depends on the amount of initial charge Q and from the distance from this charge d to a point in space, that is, a geometric location in which the value of tension is determined. Thus, we have the opportunity to characterize the electric field through its intensity.

Inverse square law

Like all formulas in physics, formulas for electric field strength can be used to algebraic solving problems (problems) of physics. Just like any other formula in its algebraic notation, you can study the formula for electric field strength. Such research contributes to more deep understanding the essence of a physical phenomenon and the characteristics of this phenomenon. One of the features of the field strength formula is that it illustrates the inverse quadratic relationship between the electric field strength and the distance to a point in space from the field source. The strength of the electric field created in the charge source Q inversely proportional to the square of the distance from the source. Otherwise they say that the desired quantity inversely proportional to the square .

The electric field strength depends on the geometric location in space, and its value decreases with increasing distance. So, for example, if the distance increases by 2 times, then the intensity will decrease by 4 times (2 2), if the distances between decrease by 2 times, then the electric field strength will increase by 4 times (2 2). If the distance increases by 3 times, then the electric field strength decreases by 9 times (3 2). If the distance increases by 4 times, then the electric field strength decreases by 16 (4 2).

Direction of the electric field strength vector

As mentioned earlier, electric field strength is a vector quantity. Unlike a scalar quantity, a vector quantity is not fully described unless its direction is specified. The magnitude of the electric field vector is calculated as the magnitude of the force at any trial charge located in an electric field.

The force acting on trial the charge can be directed either towards the charge source or directly away from it. The exact direction of the force depends on the signs of the test charge and the source of the charge, whether they have the same sign of charge (repulsion occurs) or their signs are opposite (attraction occurs). To solve the problem of the direction of the electric field vector, whether it is directed towards the source or away from the source, rules were adopted that are used by all scientists in the world. According to these rules, the direction of the vector is always from a charge with a positive polarity sign. This can be represented in the form of lines of force that come out of charges of positive signs and enter charges of negative signs.

Electrical voltage refers to the work done by an electric field to move a charge of 1 C (coulomb) from one point of a conductor to another.

How does tension arise?

All substances consist of atoms, which are a positively charged nucleus around which smaller negative electrons circle at high speed. In general, atoms are neutral because the number of electrons matches the number of protons in the nucleus.

However, if a certain number of electrons are taken away from the atoms, they will tend to attract the same number, forming a positive field around themselves. If you add electrons, then an excess of them will appear, and a negative field will appear. Potentials are formed - positive and negative.

When they interact, mutual attraction will arise.

The greater the difference - the potential difference - the stronger the electrons from the material with their excess content will be drawn to the material with their deficiency. The stronger the electric field and its voltage will be.

If you connect potentials with different charges of conductors, then an electric one will arise - a directed movement of charge carriers, tending to eliminate the potential difference. To move charges along a conductor, the electric field forces perform work, which is characterized by the concept of electric voltage.

What is it measured in?

Temperatures;

Types of voltage

Constant pressure

The voltage in the electrical network is constant when there is always a positive potential on one side and a negative potential on the other. Electric in this case has one direction and is constant.

The voltage in a direct current circuit is defined as the potential difference at its ends.

When connecting a load to a DC circuit, it is important not to mix up the contacts, otherwise the device may fail. Classic example DC voltage sources are batteries. Networks are used when there is no need to transmit energy over long distances: in all types of transport - from motorcycles to spacecraft, in military equipment, electric power and telecommunications, during emergency power supply, in industry (electrolysis, smelting in electric arc furnaces, etc.).

AC voltage

If you periodically change the polarity of the potentials, or move them in space, then the electric one will rush in the opposite direction. The number of such changes in direction over a certain time is shown by a characteristic called frequency. For example, the standard 50 means that the polarity of the voltage in the network changes 50 times per second.

Voltage in electrical networks alternating current is a time function.

The law of sinusoidal oscillations is most often used.

This happens due to what occurs in the coil of asynchronous motors due to the rotation of an electromagnet around it. If you expand the rotation in time, you get a sinusoid.

Consists of four wires - three phase and one neutral. the voltage between the neutral and phase wires is 220 V and is called phase. Between phase voltages also exist, called linear and equal to 380 V (potential difference between two phase wires). Depending on the type of connection in a three-phase network, you can get either phase voltage or linear voltage.

Placed at a given point in the field, the magnitude of this charge is:

.

From this definition it is clear why the electric field strength is sometimes called the force characteristic of the electric field (indeed, the entire difference from the force vector acting on a charged particle is only in a constant factor).

At each point in space at a given moment in time there is its own vector value (generally speaking, it is different at different points in space), thus, this is a vector field. Formally, this is expressed in the notation

representing the electric field strength as a function of spatial coordinates (and time, since it can change with time). This field, together with the field of the magnetic induction vector, is an electromagnetic field, and the laws to which it obeys are the subject of electrodynamics.

Electric field strength in SI is measured in volts per meter [V/m] or newtons per coulomb.

Electric field strength in classical electrodynamics

From the above it is clear that the electric field strength is one of the main fundamental quantities of classical electrodynamics. In this area of ​​physics, only the magnetic induction vector (together with the electric field strength vector, forming the electromagnetic field tensor) and electric charge can be called comparable in value. From some point of view, the potentials of the electromagnetic field (which together form a single electromagnetic potential) seem equally important.

• The remaining concepts and quantities of classical electrodynamics, such as electric current, current density, charge density, polarization vector, as well as auxiliary electric induction field and magnetic field strength - although quite important and significant, their significance is much less, and in fact can be considered useful and meaningful, but auxiliary quantities.

Let's give short review basic contexts of classical electrodynamics regarding electric field strength.

The force with which an electromagnetic field acts on charged particles

The total force with which the electromagnetic field (including, generally speaking, the electric and magnetic components) acts on a charged particle is expressed by the Lorentz force formula:

Where q- the electric charge of the particle, - its speed, - the magnetic induction vector (the main characteristic of the magnetic field), the oblique cross indicates the vector product. The formula is given in SI units.

As you can see, this formula is completely consistent with the definition of electric field strength given at the beginning of the article, but is more general, because also includes the action on a charged particle (if it is moving) from the magnetic field.

In this formula, the particle is assumed to be a point particle. However, this formula allows one to calculate the forces acting from the side electromagnetic field to bodies of any shape with any distribution of charges and currents - you just need to use the usual physics technique of breaking a complex body into small (mathematically - infinitely small) parts, each of which can be considered point-like and thus included in the range of applicability of the formula.

The remaining formulas used to calculate electromagnetic forces (such as, for example, the Ampere force formula) can be considered consequences of the fundamental formula of the Lorentz force, special cases of its application, etc.

However, in order for this formula to be applied (even in the simplest cases, such as calculating the force of interaction between two point charges), it is necessary to know (be able to calculate) and what the following paragraphs are devoted to.

Maxwell's equations

Together with the Lorentz force formula, a sufficient theoretical foundation for classical electrodynamics are the electromagnetic field equations, called Maxwell's equations. Their standard traditional form is four equations, three of which include the electric field strength vector:

Here - charge density, - current density, - universal constants(the equations here are written in SI units).

Here is the most fundamental and simple form Maxwell's equations - the so-called "equations for vacuum" (although, contrary to the name, they are quite applicable to describe the behavior of the electromagnetic field in a medium). Details about other forms of writing Maxwell's equations -.

These four equations, together with the fifth - the Lorentz force equation - are in principle sufficient to completely describe classical (that is, not quantum) electrodynamics, that is, they represent it complete laws. To solve specific real problems with their help, equations of motion of “material particles” (in classical mechanics these are Newton's laws), as well as often additional information about the specific properties of physical bodies and media involved in the consideration (their elasticity, electrical conductivity, polarizability, etc.), as well as about other forces involved in the problem (for example, gravity), but all this information is no longer within the scope of electrodynamics as such, although it often turns out to be necessary for constructing a closed system of equations that makes it possible to solve a particular problem as a whole.

"Material Equations"

Such additional formulas or equations (usually not exact, but approximate, often just empirical) that are not directly included in the field of electrodynamics, but are inevitably used in it to solve specific practical problems, called “ material equations"are, in particular:

• Law of Polarization
• in different cases, many other formulas and relationships.

Connection with potentials

The relationship between electric field strength and potentials in the general case is as follows:

where are the scalar and vector potentials. For completeness, we present here the corresponding expression for the magnetic induction vector:

In the special case of stationary (not changing with time) fields, the first equation simplifies to:

This is an expression for the relationship between the electrostatic field and the electrostatic potential.

Electrostatics

An important special case in electrodynamics from a practical and theoretical point of view is the case when charged bodies are stationary (for example, if the state of equilibrium is being studied) or the speed of their movement is small enough to allow approximately the use of those calculation methods that are valid for stationary bodies. This special case is dealt with by the branch of electrodynamics called electrostatics.

The field equations (Maxwell's equations) are also greatly simplified (equations with magnetic field can be excluded, and can be substituted into the equation with divergence) and are reduced to the Poisson equation:

and in areas free of charged particles - to Laplace's equation:

Considering the linearity of these equations, and therefore the applicability of the superposition principle to them, it is enough to find the field of one point unit charge in order to then find the potential or field strength created by any distribution of charges (by summing the solutions for a point charge).

Gauss's theorem

Gauss's theorem turns out to be very useful in electrostatics, the content of which boils down to integral form Maxwell's only non-trivial equation for electrostatics:

where integration is performed over any closed surface S(calculating the flux through this surface), Q- total (total) charge inside this surface.

This theorem provides an extremely simple and convenient way to calculate the electric field strength in the case when the sources have a sufficiently high symmetry, namely spherical, cylindrical or mirror + translation. In particular, the field of a point charge, sphere, cylinder, plane can be easily found in this way.

Electric field strength of a point charge

In SI units

For a point charge in electrostatics, Coulomb's law is true

. .

Historically, Coulomb's law was discovered first, although with theoretical point From a perspective, Maxwell's equations are more fundamental. From this point of view, it is their consequence. The easiest way to obtain this result is based on , taking into account the spherical symmetry of the problem: choose a surface S in the form of a sphere with a center at a point charge, take into account that the direction will be obviously radial, and the magnitude of this vector is the same everywhere on the selected sphere (so E can be taken out of the integral sign), and then, taking into account the formula for the area of ​​a sphere of radius r: , we have:

where we immediately get the answer for E.

The answer for is then obtained by integration E:

For the GHS system

The formulas and their derivation are similar, the difference from SI is only in the constants.

Electric field strength of an arbitrary charge distribution

According to the principle of superposition for the field strength of a set of discrete sources, we have:

where is each

Substituting, we get:

For a continuous distribution it is similar:

Where V- the region of space where the charges are located (non-zero charge density), or the entire space, - the radius vector of the point for which we calculate , - the radius vector of the source, running through all points of the region V when integrating, dV- element of volume. You can substitute x,y,z instead of, instead of, instead of dV.

Unit systems

In the SGS system, the electric field strength is measured in SGSE units, in the system

Definition

Tension vector– this is the force characteristic of the electric field. At a certain point in the field, the intensity is equal to the force with which the field acts on a unit positive charge placed at the specified point, while the direction of the force and the intensity coincide. The mathematical definition of tension is written as follows:

where is the force with which the electric field acts on a stationary, “test” point charge q, which is placed at the field point under consideration. In this case, it is believed that the “test” charge is small enough that it does not distort the field under study.

If the field is electrostatic, then its strength does not depend on time.

If the electric field is uniform, then its strength is the same at all points of the field.

Electric fields can be represented graphically using lines of force. Lines of force (tension lines) are lines whose tangents at each point coincide with the direction of the tension vector at that point in the field.

The principle of superposition of electric field strengths

If the field is created by several electric fields, then the strength of the resulting field is equal to the vector sum of the strengths of the individual fields:

Let us assume that the field is created by a system of point charges and their distribution is continuous, then the resulting intensity is found as:

integration in expression (3) is carried out over the entire charge distribution region.

Field strength in a dielectric

The field strength in a dielectric is equal to the vector sum of the field strengths created by free charges and bound (polarization charges):

In the event that the substance that surrounds the free charges is a homogeneous and isotropic dielectric, then the voltage is equal to:

where is the relative dielectric constant of the substance at the field point under study. Expression (5) means that for a given charge distribution, the electrostatic field strength in a homogeneous isotropic dielectric is several times less than in vacuum.

Point charge field strength

The field strength of a point charge q is equal to:

where F/m (SI system) is the electrical constant.

The relationship between tension and potential

In general, the electric field strength is related to the potential as:

where is the scalar potential, and is the vector potential.

For stationary fields, expression (7) is transformed into the formula:

Electric field strength units

The basic unit of measurement of electric field strength in the SI system is: [E]=V/m(N/C)

Examples of problem solving

Example

Exercise. What is the magnitude of the electric field strength vector at a point determined by the radius vector (in meters), if the electric field creates a positive point charge (q=1C), which lies in the XOY plane and its position is determined by the radius vector , (in meters)?

Solution. The voltage modulus of the electrostatic field that creates a point charge is determined by the formula:

r is the distance from the charge creating the field to the point at which we are looking for the field.

From formula (1.2) it follows that the module is equal to:

Substituting the initial data and the resulting distance r into (1.1), we have:

Answer.

Example

Exercise. Write down an expression for the field strength at a point determined by the radius vector if the field is created by a charge that is distributed throughout the volume V with density .

Solution. Let's make a drawing.

Let us divide the volume V into small areas with volumes and charges of these volumes, then the field strength of a point charge at point A (Fig. 1) will be equal to:

In order to find the field that creates the entire body at point A, we use the principle of superposition:

where N is the number of elementary volumes into which volume V is divided.

The charge distribution density can be expressed as:

From expression (2.3) we obtain:

Substituting the expression for the elementary charge into formula (2.2), we have:

Since the charge distribution is given as continuous, if we tend to zero, we can move from summation to integration, then: