# Electrostatics

## Basic Theoretical Information

### Electric charge and its properties

**The electric charge** is a physical quantity that characterizes the ability of particles or bodies to enter into electromagnetic interactions. The electric charge is usually designated by the letters *q* and *Q* . In the SI system, the electric charge is measured in Pendants (C). A free charge of 1 C is a huge amount of charge that is practically not found in nature. As a rule, you will have to deal with microcolourants (1 μC = 10 ^{–6} C), nanocolons (1 nC = 10 ^{–9} C) and picoculons (1 p = 10 ^{–12} C). **Electric charge has the following properties:**

**1.** Electric charge is a kind of matter.

**2. The** electric charge does not depend on the motion of the particle and on its speed.

**3.** 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.

**4.** There are two kinds of electric charges, conventionally called **positive** and **negative** .

**5.** All charges interact with each other. In this case, like charges repel each other, opposite charges attract each other. The forces of interaction of charges are central, that is, they lie on the straight line connecting the centers of charges.

**6.** There is a minimum possible (modulo) electric charge, called an **elementary charge** . Its meaning is:

*e* = 1.602177 · 10 ^{–19} Cl ≈ 1.6 · 10 ^{–19} Cl.

The electric charge of any body is always a multiple of the elementary charge:

where: *N* is an integer. Please note that the existence of a charge equal to 0.5 *e* ; 1.7 *e* ; 22.7 *e* and so on. Physical quantities that can take only a discrete (non-continuous) series of values are called **quantized** . Elementary charge e is a quantum (the smallest portion) of electric charge.

**7. The law of conservation of electric charge. **In an isolated system, the algebraic sum of the charges of all bodies remains constant:

The law of conservation of electric charge states that in a closed system of bodies the processes of creation or disappearance of charges of only one sign cannot be observed. It also follows from the charge conservation law, if two bodies of the same size and shape, having charges *q *_{1} and *q *_{2} (no matter what sign of charge), come into contact, and then dissolve back, then the charge of each of the bodies becomes equal:

From the modern point of view, charge carriers are elementary particles. All ordinary bodies are composed 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 the proton and the electron are exactly the same in modulus and equal to the elementary (that is, the minimum possible) 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 the atomic number. An atom of a given substance may lose one or more electrons, or acquire an extra electron. In these cases, the neutral atom becomes a positively or negatively charged ion. Please note that positive protons are part of the nucleus of an atom, so their number can only change during nuclear reactions. Obviously, when electrifying bodies of nuclear reactions does not occur. Therefore, in any electrical phenomena the number of protons does not change, only the number of electrons changes. So, sending a negative charge to the body means passing extra electrons to it. And the message of a positive charge, in spite of a common error, means not the addition of protons, but the taking of electrons.

Sometimes in problems the electric charge is distributed over a certain body. To describe this distribution, the following values are entered:

**1. Linear charge density. **Used to describe the charge distribution over a thread:

where: *L* is the length of the thread. Measured in C / m.

**2. Surface charge density. **Used to describe the charge distribution over the body surface:

where: *S* is the surface area of the body. Measured in C / m ^{2} .

**3. Bulk charge density. **Used to describe the distribution of charge over body volume:

where: *V* – body volume. Measured in C / m ^{3} .

Note that **the electron mass** is equal to:

*m _{e}* = 9.11 ∙

^{10–31}kg.

### Coulomb’s law

**A point charge** is a charged body whose dimensions under the conditions of this task can be neglected. Based on numerous experiments, Coulomb established the following law:

**The interaction forces of fixed point charges are directly proportional to the product of charge modules and inversely proportional to the square of the distance between them:**

where: *ε* is the dielectric constant of a medium — a dimensionless physical quantity that indicates how many times the force of electrostatic interaction in a given medium will be less than in a vacuum (that is, how many times the medium weakens the interaction). Here *k* is the coefficient in the Coulomb law, the value that determines the numerical value of the interaction force of the charges. In the SI system, its value is assumed to be:

*k* = 9 ∙ 10 ^{9} m / F.

The interaction forces of the pointless stationary charges obey the third law of Newton, and are repulsive forces from each other with the same signs of charges and forces of attraction to each other with different signs. The interaction of fixed electric charges is called **electrostatic** or Coulomb interaction. The section of electrodynamics that studies the Coulomb interaction is called **electrostatics** .

Coulomb’s law is valid for point charged bodies, uniformly charged spheres and balls. In this case, for distances *r* take the distance between the centers of spheres or balls. In practice, the Coulomb’s law is well satisfied if the size of charged bodies is much smaller than the distance between them. The coefficient *k* in the SI system is sometimes written in the form:

where: *ε *_{0} = 8.85 ∙ 10 ^{–12} F / m is the electric constant.

Experience shows that the forces of the Coulomb interaction obey the superposition principle: if a charged body interacts simultaneously with several charged bodies, then the resultant force acting on a given body is equal to the vector sum of the forces acting on this body from all other charged bodies.

**Remember also two important definitions:**

**Conductors** – substances containing free charge carriers. Inside the conductor, free movement of electrons – charge carriers is possible (an electric current can flow through the conductors). The conductors include metals, solutions and melts of electrolytes, ionized gases, plasma.

**Dielectrics (insulators)** are substances in which there are no free charge carriers. The free movement of electrons inside dielectrics is impossible (no electric current can flow through them). It is dielectrics that have some non-unit dielectric constant *ε* .

For the dielectric constant of a substance, the following is true (that is, an electric field is slightly lower):

### Electric field and its intensity

According to modern concepts, electric charges do not act on each other directly. Each charged body creates an **electric field** in the surrounding space . This field has a powerful effect on other charged bodies. The main property of the electric field is the effect on electric charges with some force. Thus, the interaction of charged bodies is carried out not by their direct action against each other, but through the electric fields surrounding the charged bodies.

The electric field surrounding a charged body can be investigated with the help of the so-called trial charge — a small point charge that does not introduce a noticeable redistribution of the charges under study. To quantify the electric field force characteristic is introduced – **the electric field strength ***E* .

The strength of the electric field is called a physical quantity equal to the ratio of the force with which the field acts on the trial charge placed at a given point of the field to the magnitude of this charge:

Electric field strength is a vector physical quantity. The direction of the intensity vector coincides at each point in space with the direction of the force acting on the positive test charge. The electric field of the fixed and unchanging with time charges is called electrostatic.

For a visual representation of the electric field using **lines of force** . These lines are drawn so that the direction of the intensity vector at each point coincides with the direction of the tangent to the field line. Power lines have the following properties.

- The power lines of the electrostatic field never intersect.
- The electrostatic field lines are always directed from positive to negative charges.
- When depicting an electric field with the help of lines of force, their density should be proportional to the modulus of the field intensity vector.
- Power lines start at a positive charge or infinity, and end at a negative or infinity. The density of the lines is greater, the greater the tension.
- At this point in space, only one line of force can pass; The electric field strength at a given point in space is uniquely defined.

An electric field is called homogeneous if the intensity vector is the same at all points of the field. For example, a uniform field creates a flat capacitor — two plates, charged by an equal in magnitude and opposite in sign charge, separated by a dielectric layer, and the distance between the plates is much smaller than the sizes of the plates.

At all points of a uniform field, a charge *q* introduced into a uniform field with a strength *E* acts on the same in magnitude and direction force, equal to *F* = *Eq* . Moreover, if the charge *q is* positive, then the direction of the force coincides with the direction of the intensity vector, and if the charge is negative, then the strength and intensity vectors are oppositely directed.

**The force lines of the Coulomb fields of** positive and negative point charges are shown in the figure:

### Principle of superposition

If using a trial charge, the electric field created by several charged bodies is investigated, then the resulting force is equal to the geometric sum of the forces acting on the trial charge from each charged body separately. Consequently, the intensity of the electric field created by the charge system at a given point in space is equal to the vector sum of the strengths of the electric fields created at the same point by the charges separately:

This property of the electric field means that the field obeys the **superposition principle** . In accordance with the Coulomb’s law, the strength of the electrostatic field created by a point charge *Q* at a distance *r* from it is equal in absolute value:

This field is called Coulomb. In the Coulomb field, the direction of the intensity vector depends on the sign of the charge *Q*: if *Q* > 0, then the intensity vector is directed from the charge, if *Q* <0, then the intensity vector is directed towards the charge. The magnitude of the strength depends on the magnitude of the charge, the medium in which the charge is located, and decreases with increasing distance.

The electric field strength, which creates a charged plane near its surface:

So, if in the task it is required to determine the field strength of the charge system, then we must act according to the following **algorithm** :

- Draw a picture.
- Draw the field strength of each charge separately at the desired point. Remember that the intensity is directed towards the negative charge and from the positive charge.
- Calculate each of the tensions using the appropriate formula.
- Add the stress vector geometrically (i.e., vector).

### Potential charge interaction energy

Electric charges interact with each other and with the electric field. Any interaction describes the potential energy. **The potential energy of interaction of two point electric charges is** calculated by the formula:

Pay attention to the absence of modules in charges. For opposite charges, the interaction energy is negative. The same formula is valid for the interaction energy of uniformly charged spheres and balls. As usual, in this case the distance r is measured between the centers of the balls or spheres. If the charges are not two, but more, then the energy of their interaction should be considered as follows: break the system of charges into all possible pairs, calculate the interaction energy of each pair and sum up all the energies for all pairs.

Tasks on this topic are solved, as well as tasks on the law of conservation of mechanical energy: first, the initial interaction energy is found, then the final one. If the task is asked to find work on the movement of charges, then it will be equal to the difference between the initial and final total interaction energy of the charges. The interaction energy can also transfer to kinetic energy or to other forms of energy. If the bodies are at a very large distance, then the energy of their interaction is set to 0.

Note: if the task requires to find the minimum or maximum distance between the bodies (particles) while moving, then this condition will be fulfilled at that moment of time when the particles move in one direction with the same speed. Therefore, the solution must begin with the recording of the law of conservation of momentum, from which this identical velocity is found. And then you should write the law of energy conservation taking into account the kinetic energy of the particles in the second case.

### Potential. Potential difference. Voltage

The electrostatic field has an important property: the **work of the electrostatic field forces when a charge moves from one point of the field to another does not depend on the shape of the trajectory, but is determined only by the position of the initial and final points and the charge value.**

The consequence of the independence of the work from the shape of the trajectory is the following statement: the work of the electrostatic field forces when the charge moves along any closed trajectory is zero.

The property of potentiality (independence of work from the shape of the trajectory) of an electrostatic field allows one to introduce the concept of potential energy of a charge in an electric field. A physical quantity equal to the ratio of the potential energy of an electric charge in an electrostatic field to the magnitude of this charge is called the **potential ***φ of the* electric field:

The potential *φ* is the energy characteristic of the electrostatic field. In the International System of Units (SI), the potential unit (and hence the potential difference, i.e. voltage) is the volt [V]. Potential is a scalar quantity.

In many problems of electrostatics when calculating the potentials for the reference point, where the values of potential energy and potential turn to zero, it is convenient to take an infinitely distant point. In this case, the concept of potential can be defined as follows: the potential of the field at a given point in space is equal to the work that the electric forces do when a single positive charge is removed from a given point to infinity.

Recalling the formula for the potential energy of the interaction of two point charges and dividing it by the value of one of the charges in accordance with the definition of potential, we obtain that the **potential ***φ of ***the point charge field ***Q* at a distance *r* from it relative to an infinitely remote point is calculated as follows

The potential calculated by this formula can be positive and negative, depending on the sign of the charge that created it. The same formula expresses the potential of the field of a uniformly charged ball (or sphere) with *r* ≥ *R* (outside the ball or sphere), where *R* is the radius of the ball, and the distance *r is* measured from the center of the ball.

For a visual representation of the electric field, along with the lines of force, **equipotential surfaces are used** . A surface, at all points of which the potential of the electric field has the same value, is called an equipotential surface or a surface of equal potential. The electric field lines are always perpendicular to the equipotential surfaces. The equipotential surfaces of the Coulomb field of a point charge are concentric spheres.

Electrical **voltage** is simply a voltage difference, i.e. The definition of electrical voltage can be given by:

In a uniform electric field, there is a relationship between field strength and voltage:

**The work of the electric field** can be calculated as the difference between the initial and final potential energy of the charge system:

The work of the electric field in the general case can also be calculated by one of the formulas:

In a uniform field when the charge moves along its lines of force, the field operation can also be calculated using the following formula:

In these formulas:

*φ*is the electric field potential.- ∆
*φ*is the potential difference. *W*is the potential energy of a charge in an external electric field.*A*– the work of the electric field on the movement of charge (charges).*q*– charge, which is moved in an external electric field.*U*is the voltage.*E*is the electric field strength.*d*or ∆*l*is the distance over which the charge moves along the lines of force.

All the previous formulas dealt specifically with the work of the electrostatic field, but if the task states that “the work must be done,” or it is referred to the “work of external forces,” then this work should be considered the same as the work of the field, but with opposite sign.

#### Principle of superposition of potential

The principle of superposition for potentials follows from the principle of superposition of the strengths of fields generated by electric charges (the sign of the field potential depends on the sign of the charge that created the field):

Notice how much easier it is to apply the principle of superposition of potential than tension. Potential is a scalar quantity that has no direction. Adding potentials is simply summing up the numerical values.

### Electric capacity Flat capacitor

When communicating to the charge conductor, there is always a certain limit, over which the body cannot be charged. To characterize the body’s ability to accumulate electrical charge, the concept of **electrical capacitance** is introduced . The capacity of a solitary conductor is the ratio of its charge to potential:

In the SI system, capacitance is measured in Farads [F]. 1 Farad – extremely large capacity. For comparison, the capacity of the entire globe is significantly less than one farad. The capacity of the conductor does not depend on its charge, nor on the potential of the body. Similarly, density does not depend on mass or on the volume of the body. Capacity depends only on the shape of the body, its size and the properties of its environment.

**The electrical capacity of a** system of two conductors is a physical quantity, defined as the ratio of the charge *q of* one of the conductors to the potential difference Δ *φ* between them:

The amount of electrical capacity of conductors depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are configurations of conductors in which the electric field is concentrated (localized) only in a certain region of space. Such systems are called **capacitors** , and the conductors that make up the capacitor are called **plates** .

The simplest capacitor is a system of two flat conducting plates arranged parallel to each other at a small distance compared to the dimensions of the plates and separated by a dielectric layer. Such a capacitor is called **flat** . The electric field of a plane capacitor is mainly localized between the plates.

Each of the charged plates of a flat capacitor creates an electric field near its surface, the strength of which is expressed by the ratio already mentioned above. Then the modulus of the strength of the final field inside a capacitor created by two plates is equal to:

Outside of the capacitor, the electric fields of the two plates are directed in different directions, and therefore the resulting electrostatic field is *E* = 0. The **capacitance of the flat capacitor** can be calculated by the formula

Thus, the electrical capacitance of a flat capacitor is directly proportional to the area of the plates (plates) and inversely proportional to the distance between them. If the space between the plates is filled with a dielectric, the capacitance of the capacitor increases by a factor of *ε* . Note that *S* in this formula is the area of only one capacitor plate. When the problem is referred to as “area of the plates”, they mean precisely this quantity. It is never necessary to multiply or divide by 2.

Once again we give the formula for **charging the capacitor** . Under the charge of the capacitor understand only the charge of its positive lining:

**The force of attraction of the capacitor plates. **The force acting on each plate is determined not by the full field of the capacitor, but by the field created by the opposite plate (the plate itself does not act). The strength of this field is equal to half the strength of the total field, and the strength of the interaction of the plates:

**Condenser energy. **It is also called the energy of the electric field inside the capacitor. Experience shows that a charged capacitor contains a supply of energy. The energy of a charged capacitor is equal to the work of external forces, which must be expended to charge the capacitor. There are three equivalent forms for writing a formula for the energy of a capacitor (they follow one of the other if we use the relation *q* = *CU* ):

Pay special attention to the phrase: “The capacitor is connected to the source.” This means that the voltage across the capacitor does not change. And the phrase “The capacitor was charged and disconnected from the source” means that the charge of the capacitor will not change.

#### Electric field energy

Electrical energy should be considered as potential energy stored in a charged capacitor. According to modern concepts, the electric energy of a capacitor is localized in the space between the capacitor plates, that is, in an electric field. Therefore, it is called the electric field energy. The energy of charged bodies is concentrated in space, in which there is an electric field, i.e. You can talk about the energy of the electric field. For example, for a capacitor, energy is concentrated in the space between its plates. Thus, it makes sense to introduce a new physical characteristic – the bulk density of the electric field energy. Using the example of a flat capacitor, one can obtain the following formula for the bulk energy density (or energy per unit volume of the electric field):

### Capacitor connections

**A Parallel connection of capacitors** – to increase capacity. Capacitors are connected by the same charged plates, as if increasing the area of equally charged plates. The voltage on all capacitors is the same, the total charge is equal to the sum of the charges of each of the capacitors, and the total capacity is also equal to the sum of the capacitors of all the capacitors connected in parallel. We write the formulas for parallel connection of capacitors:

When **connecting capacitors in series, the** total capacitance of the capacitor bank is always less than the capacity of the smallest capacitor entering the battery. A series connection is used to increase the breakdown voltage of capacitors. We write the formulas for the serial connection of capacitors. The total capacitance of series-connected capacitors is found from the relationship:

From the law of conservation of charge, it follows that the charges on the adjacent plates are equal:

The voltage is equal to the sum of the voltages on the individual capacitors.

For two series-connected capacitors, the formula above will give us the following expression for the total capacity:

For *N* identical series-connected capacitors:

### Conductive sphere

**The field strength inside the charged conductor is zero. **Otherwise, the free charges inside the conductor would be acted upon by an electric force, which would force these charges to move inside the conductor. This movement, in turn, would lead to heating of the charged conductor, which in fact does not occur.

The fact that there is no electric field inside the conductor can be understood in another way: if it were, then the charged particles would move again, and they would move precisely so as to reduce this field to zero by its own field, since in general, they would not want to move, because any system tends to equilibrium. Sooner or later, all moving charges would stop at exactly that place so that the field inside the conductor becomes equal to zero.

On the surface of the conductor the electric field strength is maximum. The magnitude of the electric field of a charged ball outside it decreases with distance from the conductor and is calculated using a formula similar to the formulas for the field strength of a point charge, in which the distances are measured from the center of the ball.

Since the field strength inside a charged conductor is zero, the potential at all points inside and on the surface of the conductor is the same (only in this case, the potential difference, and hence the strength is zero). **The potential inside a charged ball is equal to the potential on the surface. **The potential outside the ball is calculated by a formula similar to the formulas for the point charge potential, in which the distances are measured from the center of the ball.

**Electrical capacity of the ball of** radius *R* :

If the ball is surrounded by a dielectric, then:

### Properties of the conductor in the electric field

- Inside the conductor, the field strength is always zero.
- The potential inside the conductor at all points is the same and equal to the potential of the surface of the conductor. When the problem says that “the conductor is charged to the potential … B”, then it is the surface potential that is meant.
- Outside of the conductor near its surface, the field strength is always perpendicular to the surface.
- If the conductor is charged, it is distributed over a very thin layer near the surface of the conductor (they usually say that the entire charge of the conductor is distributed on its surface). This is easily explained: the fact is that when we communicate the charge to the body, we give it charge carriers of the same sign, i.e. charges of the same name that repel each other. So they will strive to scatter from each other to the maximum distance of all possible, i.e. accumulate at the very edges of the conductor. As a result, if the core is removed from the conductor, its electrostatic properties will not change at all.
- Outside the conductor, the field strength is greater, the more curved the surface of the conductor. The maximum value of tension is reached near the tips and sharp kinks of the conductor surface.

### Tips for solving complex problems

**1. Grounding** something means connecting the conductor of a given object to the Earth. In this case, the potentials of the Earth and the existing object are aligned, and the necessary charges for this run across the conductor from the Earth to the object or vice versa. In this case, it is necessary to take into account several factors that follow from the fact that the Earth is disproportionately larger than any object that is not located there:

- The total charge of the Earth is conventionally equal to zero, therefore its potential is also equal to zero, and it will remain equal to zero after the connection of the object with the Earth. In short, grounding means zeroing the potential of the object.
- To reset the potential (and hence the object’s own charge, which could have been both positive and negative), the object will have to either accept or give the Earth some (possibly very large) charge, and the Earth will always be able to provide such an opportunity.

**2.** We repeat once again: the distance between repulsive bodies is minimal at the moment when their velocities become equal in magnitude and are directed in one direction (the relative velocity of the charges is zero). At this moment, the potential energy of the interaction of charges is maximum. The distance between the attracting bodies is maximal, also at the moment of equality of speeds directed in one direction.

**3.** If the problem has a system consisting of a large number of charges, then it is necessary to consider and describe the forces acting on a charge that is not in the center of symmetry.