>>Physics: Field lines electric field. Field strength of a charged ball
The electric field does not affect the senses. We don't see him.
However, we can get some idea of the field distribution if we draw the field strength vectors at several points in space ( Fig.14.9, left). The picture will be more clear if you draw continuous lines, the tangents to which at each point through which they pass coincide in direction with the tension vectors. These lines are called electric field lines or tension lines (Fig.14.9, right).
Direction power lines allows you to determine the direction of the intensity vector at different points of the field, and the density (number of lines per unit area) of the field lines shows where the field strength is greater. So, in Figures 14.10-14.13 the density of field lines at points A more than points IN. Obviously, 
.
One should not think that tension lines actually exist like stretched elastic threads or cords, as Faraday himself assumed. Tension lines only help to visualize the distribution of the field in space. They are no more real than the meridians and parallels on globe.
However, field lines can be made visible. If elongated crystals of an insulator (for example, quinine) are mixed well in a viscous liquid (for example, castor oil) and charged bodies are placed there, then near these bodies the crystals will line up in chains along the lines of tension.
The figures show examples of tension lines: a positively charged ball (see. Fig.14.10); two differently charged balls (see. Fig.14.11); two similarly charged balls (see. Fig.14.12); two plates whose charges are equal in magnitude and opposite in sign (see. Fig.14.13). Last example especially Figure 14.13 shows that in the space between the plates closer to the middle, the lines of force are parallel: the electric field here is the same at all points.
An electric field whose strength is the same at all points in space is called homogeneous. IN limited area space, the electric field can be considered approximately uniform if the field strength inside this region changes slightly.
A uniform electric field is depicted parallel lines located on equal distances from each other.
The electric field lines are not closed; they begin on positive charges and end on negative ones. The lines of force are continuous and do not intersect, since intersection would mean the absence of a specific direction of the electric field strength at a given point.
Field of a charged ball. Let us now consider the question of the electric field of a charged conducting ball of radius R. Charge q evenly distributed over the surface of the ball. The electric field lines, as follows from symmetry considerations, are directed along the extensions of the radii of the ball ( Fig. 14.14, a).
Pay attention! Power the lines outside the ball are distributed in space in exactly the same way as the field lines point charge (Fig.14.14, b). If the patterns of field lines coincide, then we can expect that the field strengths also coincide. Therefore, at a distance r>R from the center of the ball, the field strength is determined by the same formula (14.9) as the field strength of a point charge placed at the center of the sphere:
The picture of the field lines clearly shows how the electric field strength is directed at different points in space. By changing the density of the lines, one can judge the change in the modulus of the field strength when moving from point to point.
???
1. What are electric field lines called?
2. In all cases, does the trajectory of a charged particle coincide with the field line?
3. Can lines of force intersect?
4. What is the field strength of a charged conducting ball?
G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade
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Gauss's theorem.
Tension vector flow through a closed contour with area S is the product of the projection of the tension vector on the normal to the contour by the area of the contour: .
The flux of the tension vector through an arbitrary closed surface is equal to the algebraic sum of the charges located inside this surface, divided by the electrical constant: 
.
Field strength of a point charge.
To determine the intensity, we draw a spherical surface S of radius r with the center coinciding with the charge and use Gauss’s theorem. Since there is only one charge q inside the indicated area, then according to the indicated theorem we obtain the equality: (1), where E n is the normal component of the electric field strength. For reasons of symmetry, the normal component must be equal to the tension itself and constant for all points of the spherical surface, therefore E=E n =const. Therefore, it can be taken out as a sum sign. Then equality (1) will take the form 
, which was obtained from Coulomb’s law and the determination of the electric field strength.
Electric field of a charged sphere
If the sphere is conductive, then all the charge is on the surface. Let us consider two regions I – inside a sphere of radius R with charge q and outside the sphere region II.
In region II R£r 2 we draw a spherical surface S 2 with radius r 2 and use Gauss’s theorem:
(2), Þ 
- the field strength outside the sphere is calculated using the same formula as the field strength of a point charge.
Electric field of a charged ball
The charge is uniformly distributed throughout the entire volume of the ball, so we introduce the concept of volumetric charge density: . Let us consider two regions I – inside a sphere of radius R with charge q and outside the sphere region II.
To determine the tension in region I, we draw a spherical surface S 1 with radius r 1 (0
In region II R £ r 2 we draw a spherical surface S 2 with radius r 2 and use Gauss’s theorem:
(2), Þ - the field strength outside the ball is calculated using the same formula as the field strength of a point charge.
« Physics - 10th grade"
What do field lines show?
What are they used for?
Field strength of a point charge.
Let's find the electric field strength created by a point charge q 0 . According to Coulomb's law, this charge will act on the positive charge q with a force
The modulus of the field strength of a point charge q 0 at a distance r from it is equal to:
The intensity vector at any point of the electric field is directed along the straight line connecting this point and the charge (Fig. 14.14), and coincides with the force acting on a point positive charge placed at a given point.
The electric field lines of a point charge, as follows from symmetry considerations, are directed along the radial lines (Fig. 14.15, a).
Field of a charged ball.
Let us now consider the question of the electric field of a charged conducting ball of radius R. Charge q is uniformly distributed over the surface of the ball. The electric field lines, also for reasons of symmetry, are directed along the extensions of the radii of the ball (Fig. 14.15, b).
The distribution in space of the electric field lines of a ball with a charge q at distances r ≥ R from the center of the ball is similar to the distribution of the field lines of a point charge q (see Fig. 14.15, a). Consequently, at a distance r ≥ R from the center of the ball, the field strength is determined by the same formula (14.9) as the field strength of a point charge placed at the center of the sphere:
Inside the conducting ball (r< R) напряженность поля равна нулю.
The principle of superposition of fields.
If several forces act on a body, then, according to the laws of mechanics, the resulting force is equal to the geometric sum of these forces:
1 + 2 + ... .
Electric charges are acted upon by forces from the electric field. If, when fields from several charges are superimposed, these fields do not have any influence on each other, then the resulting force from all fields must be equal to the geometric sum of the forces from each field. Experience shows that this is exactly what happens in reality. This means that the field strengths add up geometrically.
This is the principle of field superposition
If at a given point in space various charged particles create electric fields whose strengths are 1, 2, 3, etc., then the resulting field strength at this point is equal to the sum of the strengths of these fields:
= 1 + 2 + 3 + ... . (14.11)
The field strength created by a single charge is determined as if there were no other charges creating the field.
According to the principle of field superposition, to find the field strength of a system of charged particles at any point, it is enough to know expression (14.9) for the field strength of a point charge.
To determine the direction of the field strength vectors of individual charges, we mentally place a positive charge at the selected point.
Figure 14.16 shows how the field strength at point A created by two point charges q 1 and q 2 is determined.
Source: “Physics - 10th grade”, 2014, textbook Myakishev, Bukhovtsev, Sotsky
Electrostatics - Physics, textbook for grade 10 - Cool physics
What is electrodynamics ---
Let us determine the electric field strength of charged bodies of simple shape: a sphere and a plane. Many bodies in nature and technology have an approximately spherical shape: atomic nuclei, raindrops, planets, etc. Flat surfaces are also common. In addition, a small area of any surface can be approximately considered flat.
Ball field. Consider a charged conducting ball of radius. The charge is uniformly distributed over the surface of the ball. The electric field lines, as follows from symmetry considerations, are directed along the extensions of the radii of the ball (Fig. 112).
Please note: the field lines outside the ball are distributed in space in exactly the same way as the field lines of a point charge (Fig. 113). If the patterns of field lines coincide, then we can expect that the field strengths also coincide. Therefore, at a distance from the center of the ball, the field strength
is determined by the same formula (8.11) as the field strength of a point charge placed at the center of the sphere:
Rigorous calculations also lead to this result.
Inside the conducting ball, the field strength is zero.
Plane field. The distribution of electric charge on the surface of a charged body is characterized by a special value - the surface charge density o. Surface charge density is the ratio of charge to the surface area over which it is distributed. If the charge is uniformly distributed over a surface whose area is 5, then
Name of the unit of surface charge density
From symmetry considerations, it is obvious that the electric field lines of an infinite uniformly charged plane are straight lines perpendicular to the plane (Fig. 114). The field of an infinite plane is a homogeneous field, that is, at all points in space, regardless of the distance to the plane, the field strength is the same. It is determined by the surface charge density.
To find the dependence of the field strength on the surface charge density o, you can use a method often used in physics, based on knowledge of the names of physical quantities. The unit of electric field strength is called a unit of surface charge density
In order to obtain the correct name for the unit of field strength in this case, we must assume that
An infinite plane charged with a surface charge density: to calculate the electric field strength created by an infinite plane, we select a cylinder in space, the axis of which is perpendicular to the charged plane, and the bases are parallel to it, and one of the bases passes through the field point of interest to us. According to Gauss's theorem, the flux of the electric field strength vector through a closed surface is equal to:
Ф=, on the other hand it is also: Ф=E
Let's equate the right sides of the equations:
Let us express = - through the surface charge density and find the electric field strength:
Let us find the electric field strength between oppositely charged plates with the same surface density:
(3)
Let's find the field outside the plates:
; ; 
(4)
Field strength of a charged sphere
(1)
Ф= (2) Gaussian point
for r< R
; , because (there are no charges inside the sphere)
For r = R
( ; ; 
) 
For r > R
Field strength created by a ball charged uniformly throughout its volume
Volume charge density,
distributed over the ball:
For r< R
( ; Ф= ) 
For r = R
For r > R
WORK OF THE ELECTROSTATIC FIELD TO MOVE A CHARGE
Electrostatic field- email field of a stationary charge.
Fel, acting on the charge, moves it, performing work.
In a uniform electric field Fel = qE is a constant value
Work field (el. force) does not depend on the shape of the trajectory and on a closed trajectory = zero.
If in the electrostatic field of a point charge Q another point charge Q 0 moves from point 1 to point 2 along any trajectory (Fig. 1), then the force that is applied to the charge does some work. The work done by force F on an elementary displacement dl is equal to Since d l/cosα=dr, then  The work when moving a charge Q 0 from point 1 to point 2 (1) does not depend on the trajectory of movement, but is determined only by the positions of the initial 1 and final 2 points. This means that the electrostatic field of a point charge is potential, and the electrostatic forces are conservative. From formula (1) it is clear that the work that is done when an electric charge moves in an external electrostatic field along an arbitrary closed path L is equal to zero, i.e. (2) If we take a single point positive charge as a charge that is moved in an electrostatic field, then the elementary work of field forces along the path dl is equal to Edl = E l d l, where E l= Ecosα - projection of vector E onto the direction of elementary displacement. Then formula (2) can be represented as  (3) Integral  is called the circulation of the tension vector. This means that the circulation of the tension vector electrostatic field along any closed contour is zero. A force field that has property (3) is called potential. From the fact that the circulation of vector E is equal to zero, it follows that the lines of electrostatic field strength cannot be closed; they necessarily begin and end on charges (positive or negative) or go to infinity. Formula (3) is valid only for the electrostatic field. Subsequently, it will be shown that in the case of a field of moving charges, condition (3) is not true (for it, the circulation of the intensity vector is nonzero). |
Circulation theorem for the electrostatic field.
Since the electrostatic field is central, the forces acting on the charge in such a field are conservative. Since it represents the elementary work that field forces produce on a unit charge, the work of conservative forces on a closed loop is equal to
Potential
The "charge - electrostatic field" or "charge - charge" system has potential energy, just as the "gravitational field - body" system has potential energy.
A physical scalar quantity characterizing the energy state of the field is called potential a given point in the field. A charge q is placed in a field, it has potential energy W. Potential is a characteristic of an electrostatic field.
Let's remember potential energy in mechanics. Potential energy is zero when the body is on the ground. And when a body is raised to a certain height, it is said that the body has potential energy.
Regarding potential energy in electricity, there is no zero level of potential energy. It is chosen randomly. Therefore, potential is a relative physical quantity.
Potential field energy is the work done by the electrostatic force when moving a charge from a given point in the field to a point with zero potential.
Let's consider special case, when an electrostatic field is created by an electric charge Q. To study the potential of such a field, there is no need to introduce a charge q into it. You can calculate the potential of any point in such a field located at a distance r from the charge Q.
The dielectric constant of the medium has a known value (tabular) and characterizes the medium in which the field exists. For air it is equal to unity.
Potential difference
The work done by a field to move a charge from one point to another is called potential difference
This formula can be presented in another form
Superposition principle
The potential of a field created by several charges is equal to the algebraic (taking into account the sign of the potential) sum of the potentials of the fields of each field separately
This is the energy of a system of stationary point charges, the energy of a solitary charged conductor and the energy of a charged capacitor.
If there is a system of two charged conductors (capacitor), then the total energy of the system is equal to the sum of the own potential energies of the conductors and the energy of their interaction:
Electrostatic field energy system of point charges is equal to: 
Uniformly charged plane.
The electric field strength created by an infinite plane charged with a surface charge density can be calculated using Gauss's theorem.
From the symmetry conditions it follows that the vector E everywhere perpendicular to the plane. In addition, at points symmetric relative to the plane, the vector E will be the same in size and opposite in direction.
As a closed surface, we choose a cylinder whose axis is perpendicular to the plane, and whose bases are located symmetrically relative to the plane, as shown in the figure.
Since the lines of tension are parallel to the generatrices of the side surface of the cylinder, the flow through the side surface is zero. Therefore the vector flow E through the surface of the cylinder
,
where is the area of the base of the cylinder. The cylinder cuts a charge out of the plane. If the plane is in a homogeneous isotropic medium with relative dielectric constant, then
When the field strength does not depend on the distance between the planes, such a field is called uniform. Dependency graph E (x) for a plane.
Potential difference between two points located at a distance R 1 and R 2 from the charged plane is equal to
Example 2. Two uniformly charged planes.
Let's calculate the electric field strength created by two infinite planes. The electric charge is distributed uniformly with surface densities and . We find the field strength as a superposition of the field strengths of each of the planes. The electric field is nonzero only in the space between the planes and is equal to .
Potential difference between planes 
, Where d- distance between planes.
The results obtained can be used for an approximate calculation of the fields created by flat plates of finite dimensions if the distances between them are much less than their linear dimensions. Noticeable errors in such calculations appear when considering fields near the edges of the plates. Dependency graph E (x) for two planes.
Example 3. Thin charged rod.
To calculate the electric field strength created by a very long rod charged with a linear charge density, we use Gauss's theorem.
At sufficiently large distances from the ends of the rod, the electric field intensity lines are directed radially from the axis of the rod and lie in planes perpendicular to this axis. At all points equidistant from the axis of the rod, the numerical values of the tension are the same if the rod is in a homogeneous isotropic medium with a relative dielectric
permeability
To calculate the field strength at an arbitrary point located at a distance r from the axis of the rod, draw a cylindrical surface through this point
(see picture). The radius of this cylinder is r, and its height h.
The fluxes of the tension vector through the upper and lower bases of the cylinder will be equal to zero, since the lines of force do not have components normal to the surfaces of these bases. At all points on the lateral surface of the cylinder
E= const.
Therefore, the total flow of the vector E through the surface of the cylinder will be equal to
,
According to Gauss's theorem, the flux of the vector E equal to the algebraic sum of the electric charges located inside the surface (in this case, the cylinder) divided by the product of the electrical constant and the relative dielectric constant of the medium
where is the charge of that part of the rod that is inside the cylinder. Therefore, the electric field strength
Electric field potential difference between two points located at distances R 1 and R 2 from the axis of the rod, we find using the relationship between the intensity and potential of the electric field. Since the field strength changes only in the radial direction, then
Example 4. Charged spherical surface.
The electric field created by a spherical surface over which an electric charge with surface density is uniformly distributed has a centrally symmetrical character.
The tension lines are directed along radii from the center of the sphere, and the magnitude of the vector E depends only on the distance r from the center of the sphere. To calculate the field, we select a closed spherical surface of radius r.
When r o
The field strength is zero, since there is no charge inside the sphere.
For r > R (outside the sphere), according to Gauss’s theorem
,
where is relative permittivity environment surrounding the sphere.
.
The intensity decreases according to the same law as the field strength of a point charge, i.e. according to the law.
When r o
For r > R (outside the sphere) 
.
Dependency graph E (r) for a sphere.
Example 5. A volume-charged dielectric ball.
If the ball has radius R made of a homogeneous isotropic dielectric with relative permeability is uniformly charged throughout the volume with density , then the electric field it creates is also centrally symmetrical.
As in the previous case, we choose a closed surface to calculate the vector flux E in the form of a concentric sphere, the radius of which r can vary from 0 to .
At r < R vector flow E through this surface will be determined by the charge
So
At r < R(inside the ball) 
.
Inside the ball, the tension increases in direct proportion to the distance from the center of the ball. Outside the ball (at r > R) in a medium with dielectric constant , flux vector E through the surface will be determined by the charge.
When r o >R o (outside the ball) 
.
At the “ball - environment” boundary, the electric field strength changes abruptly, the magnitude of which depends on the ratio of the dielectric constants of the ball and the environment. Dependency graph E (r) for ball ().
Outside the ball ( r > R) the electric field potential changes according to the law
.
Inside the ball ( r < R) the potential is described by the expression
In conclusion, we present expressions for calculating the field strengths of charged bodies, various shapes
| Potential difference | |
 | |
| Voltage- the difference in potential values at the initial and final points of the trajectory. Voltage is numerically equal to the work of the electrostatic field when a unit positive charge moves along the lines of force of this field. The potential difference (voltage) is independent of the selection coordinate systems! | |
Unit of potential difference  The voltage is 1 V if, when moving a positive charge of 1 C along the lines of force, the field does 1 J of work. |
Conductor- This solid, which contains “ free electrons”, moving within the body.
Metal conductors are generally neutral: they contain equal amounts of negative and positive charges. Positively charged are the ions in the nodes crystal lattice, negative - electrons moving freely along the conductor. When a conductor is given an excess amount of electrons, it becomes charged negatively, but if a certain number of electrons are “taken” from the conductor, it becomes charged positively.
The excess charge is distributed only along outer surface conductor.
1 . The field strength at any point inside the conductor is zero.
2 . The vector on the surface of the conductor is directed normal to each point on the surface of the conductor.
From the fact that the surface of the conductor is equipotential it follows that directly at this surface the field is directed normal to it at each point (condition 2 ). If this were not so, then under the action of the tangential component the charges would begin to move along the surface of the conductor. those. equilibrium of charges on a conductor would be impossible.
From 1 it follows that since
There are no excess charges inside the conductor.
Charges are distributed only on the surface of the conductor with a certain density s and are located in a very thin surface layer (its thickness is about one or two interatomic distances).
Charge density- this is the amount of charge per unit length, area or volume, thus defining linear, surface and bulk density charge, which are measured in the SI system: in Coulombs per meter [C/m], in Coulombs per square meter[C/m²] and in Coulombs per cubic meter[C/m³], respectively. Unlike the density of matter, charge density can have both positive and negative values, this is due to the fact that there are positive and negative charges.
General task electrostatics
Tension vector,
by Gauss's theorem 
- Poisson's equation.
In the case where there are no charges between the conductors, we get
- Laplace's equation.
Let them be known boundary conditions on conductor surfaces: values 
; Then this task has the only solution according to uniqueness theorem.
When solving the problem, the value is determined and then the field between the conductors is determined by the distribution of charges on the conductors (according to the voltage vector at the surface).
Let's look at an example. Let's find the voltage in the empty cavity of the conductor.
The potential in the cavity satisfies Laplace's equation;
potential on the walls of the conductor.
The solution to Laplace's equation in this case is trivial, and by the uniqueness theorem there are no other solutions
, i.e. there is no field in the conductor cavity.
Poisson's equation- elliptical differential equation in partial derivatives, which, among other things, describes
· electrostatic field,
· stationary temperature field,
· pressure field,
· velocity potential field in hydrodynamics.
It is named after the famous French physicist and mathematician Simeon Denis Poisson.
This equation looks like:
where is the Laplace operator or Laplacian, and is real or complex function on some variety.
In three dimensions Cartesian system coordinates the equation takes the form:
In the Cartesian coordinate system, the Laplace operator is written in the form and the Poisson equation takes the form:
If f tends to zero, then the Poisson equation turns into the Laplace equation (the Laplace equation is a special case of the Poisson equation):
Poisson's equation can be solved using the Green's function; see, for example, the article Screened Poisson's equation. Eat various methods to receive numerical solutions. For example, an iterative algorithm is used - the “relaxation method”.
| We will consider a solitary conductor, i.e. a conductor significantly removed from other conductors, bodies and charges. Its potential, as is known, is directly proportional to the charge of the conductor. It is known from experience that different conductors, although equally charged, have different potentials. Therefore, for a solitary conductor we can write Quantity (1) is called the electrical capacity (or simply capacitance) of an solitary conductor. The capacitance of an isolated conductor is determined by the charge, the communication of which to the conductor changes its potential by one. The capacitance of a solitary conductor depends on its size and shape, but does not depend on the material, shape and size of the cavities inside the conductor, as well as its state of aggregation. The reason for this is that excess charges are distributed on the outer surface of the conductor. Capacitance also does not depend on the charge of the conductor or its potential. The unit of electrical capacity is the farad (F): 1 F is the capacity of an isolated conductor whose potential changes by 1 V when a charge of 1 C is imparted to it. According to the formula for the potential of a point charge, the potential of a solitary ball of radius R, which is located at homogeneous environment with dielectric constant ε, is equal to Applying formula (1), we find that the capacity of the ball (2) From this it follows that a solitary ball located in a vacuum and having a radius R=C/(4πε 0)≈9 10 would have a capacity of 1 F 6 km, which is approximately 1400 times greater than radius Earth (electric capacity of the Earth C≈0.7 mF). Therefore, farad is quite large value, therefore in practice they are used submultiples- millifarad (mF), microfarad (uF), nanofarad (nF), picofarad (pF). From formula (2) it also follows that the unit of the electrical constant ε 0 is farad per meter (F/m) (see (78.3)). |
Capacitor(from lat. condensare- “compact”, “thicken”) - a two-terminal network with a certain capacitance value and low ohmic conductivity; a device for accumulating charge and energy of an electric field. A capacitor is a passive electronic component. Typically consists of two plate-shaped electrodes (called linings), separated by a dielectric whose thickness is small compared to the size of the plates.
Capacity
The main characteristic of a capacitor is its capacity, characterizing the capacitor’s ability to accumulate electrical charge. The designation of a capacitor indicates the value of the nominal capacitance, while the actual capacitance can vary significantly depending on many factors. The actual capacitance of the capacitor determines its electrical properties. Thus, according to the definition of capacitance, the charge on the plate is proportional to the voltage between the plates ( q = CU). Typical capacitance values range from units of picofarads to thousands of microfarads. However, there are capacitors (ionistors) with a capacity of up to tens of farads.
Capacity flat capacitor, consisting of two parallel metal plates with an area S each located at a distance d from each other, in the SI system is expressed by the formula: , where is the relative dielectric constant of the medium filling the space between the plates (in a vacuum equal to unity), is the electrical constant, numerically equal to 8.854187817·10 −12 F/m. This formula is valid only when d much smaller than the linear dimensions of the plates.
To obtain large capacities, capacitors are connected in parallel. In this case, the voltage between the plates of all capacitors is the same. Total battery capacity parallel of connected capacitors is equal to the sum of the capacitances of all capacitors included in the battery.
If all parallel-connected capacitors have the same distance between the plates and the dielectric properties, then these capacitors can be represented as one large capacitor, divided into fragments of a smaller area.
When capacitors are connected in series, the charges of all capacitors are the same, since they are supplied from the power source only to the external electrodes, and on the internal electrodes they are obtained only due to the separation of charges that previously neutralized each other. Total battery capacity sequentially connected capacitors is equal to
Or 
This capacity is always less than the minimum capacity of the capacitor included in the battery. However, with a series connection, the possibility of breakdown of capacitors is reduced, since each capacitor accounts for only part of the potential difference of the voltage source.
If the area of the plates of all capacitors connected in series is the same, then these capacitors can be represented as one large capacitor, between the plates of which there is a stack of dielectric plates of all the capacitors that make it up.
[edit]Specific capacity
Capacitors are also characterized by specific capacitance - the ratio of capacitance to the volume (or mass) of the dielectric. Maximum value The specific capacitance is achieved with a minimum thickness of the dielectric, but at the same time its breakdown voltage decreases.
IN electrical circuits various methods of connecting capacitors. Connection of capacitors can be produced: sequentially, parallel And series-parallel(the latter is sometimes called a mixed connection of capacitors). Existing types The capacitor connections are shown in Figure 1.
Figure 1. Methods for connecting capacitors.
