Good day to all. In the last article I talked about the magnetic field and dwelled a little on its parameters. This article continues the topic of the magnetic field and is devoted to such a parameter as magnetic induction. To simplify the topic, I will talk about the magnetic field in a vacuum, since different substances have different magnetic properties, and as a result, it is necessary to take their properties into account.
Biot–Savart–Laplace law
As a result of studying the magnetic fields created by electric current, researchers came to the following conclusions:
- magnetic induction created by electric current is proportional to the strength of the current;
- magnetic induction depends on the shape and size of the conductor through which the electric current flows;
- magnetic induction at any point in the magnetic field depends on the location of this point in relation to the current-carrying conductor.
The French scientists Biot and Savard, who came to such conclusions, turned to the great mathematician P. Laplace to generalize and derive the basic law of magnetic induction. He hypothesized that the induction at any point of the magnetic field created by a current-carrying conductor can be represented as the sum of the magnetic inductions of elementary magnetic fields that are created by an elementary section of a current-carrying conductor. This hypothesis became the law of magnetic induction, called Biot-Savart-Laplace law. To consider this law, let us depict a current-carrying conductor and the magnetic induction it creates
Magnetic induction dB created by an elementary section of a conductor dl.
Then magnetic induction dB elementary magnetic field that is created by a section of a conductor dl, with current I at an arbitrary point R will be determined by the following expression
where I is the current flowing through the conductor,
r is the radius vector drawn from the conductor element to the magnetic field point,
dl is the minimum conductor element that creates induction dB,
k – proportionality coefficient, depending on the reference system, in SI k = μ 0 /(4π)
Because is a vector product, then the final expression for the elementary magnetic induction will look like this
Thus, this expression allows us to find the magnetic induction of the magnetic field, which is created by a conductor with a current of arbitrary shape and size by integrating the right side of the expression
where the symbol l indicates that integration occurs along the entire length of the conductor.
Magnetic induction of a straight conductor
As you know, the simplest magnetic field creates a straight conductor through which electric current flows. As I already said in the previous article, the lines of force of a given magnetic field are concentric circles located around the conductor.
To determine magnetic induction IN straight wire at a point R Let us introduce some notation. Since the point R is at a distance b from the wire, then the distance from any point on the wire to the point R is defined as r = b/sinα. Then the shortest length of the conductor dl can be calculated from the following expression
As a result, the Biot–Savart–Laplace law for a straight wire of infinite length will have the form
where I is the current flowing through the wire,
b is the distance from the center of the wire to the point at which the magnetic induction is calculated.
Now we simply integrate the resulting expression over dα ranging from 0 to π.
Thus, the final expression for the magnetic induction of a straight wire of infinite length will have the form
I – current flowing through the wire,
b is the distance from the center of the conductor to the point at which the induction is measured.
Magnetic induction of the ring
The induction of a straight wire has a small value and decreases with distance from the conductor, therefore it is practically not used in practical devices. The most widely used magnetic fields are those created by a wire wound around a frame. Therefore, such fields are called magnetic fields of circular current. The simplest such magnetic field is possessed by an electric current flowing through a conductor, which has the shape of a circle of radius R.
In this case, two cases are of practical interest: the magnetic field at the center of the circle and the magnetic field at point P, which lies on the axis of the circle. Let's consider the first case.
In this case, each current element dl creates an elementary magnetic induction dB in the center of the circle, which is perpendicular to the contour plane, then the Biot-Savart-Laplace law will have the form
All that remains is to integrate the resulting expression over the entire length of the circle
where μ 0 is the magnetic constant, μ 0 = 4π 10 -7 H/m,
I – current strength in the conductor,
R is the radius of the circle into which the conductor is rolled.
Let's consider the second case, when the point at which the magnetic induction is calculated lies on the straight line X, which is perpendicular to the plane limited by the circular current.
In this case, induction at the point R will be the sum of elementary inductions dB X, which in turn is a projection onto the axis X elementary induction dB
Applying the Biot-Savart-Laplace law, we calculate the value of magnetic induction
Now let’s integrate this expression over the entire length of the circle
where μ 0 is the magnetic constant, μ 0 = 4π 10 -7 H/m,
I – current strength in the conductor,
R is the radius of the circle into which the conductor is rolled,
x is the distance from the point at which the magnetic induction is calculated to the center of the circle.
As can be seen from the formula for x = 0, the resulting expression transforms into the formula for magnetic induction at the center of the circular current.
Circulation of the magnetic induction vector
To calculate the magnetic induction of simple magnetic fields, the Biot-Savart-Laplace law is sufficient. However, with more complex magnetic fields, for example, the magnetic field of a solenoid or toroid, the number of calculations and the cumbersomeness of the formulas will increase significantly. To simplify calculations, the concept of circulation of the magnetic induction vector is introduced.
Let's imagine some contour l, which is perpendicular to the current I. At any point R of this circuit, magnetic induction IN directed tangentially to this contour. Then the product of vectors dl And IN is described by the following expression
Since the angle dφ small enough, then the vectors dl B defined as arc length
Thus, knowing the magnetic induction of a straight conductor at a given point, we can derive an expression for the circulation of the magnetic induction vector
Now it remains to integrate the resulting expression over the entire length of the contour
In our case, the magnetic induction vector circulates around one current, but in the case of several currents, the expression for the circulation of magnetic induction turns into the law of total current, which states:
The circulation of the magnetic induction vector in a closed loop is proportional to the algebraic sum of the currents that the given loop covers.
Magnetic field of solenoid and toroid
Using the law of total current and circulation of the magnetic induction vector, it is quite easy to determine the magnetic induction of such complex magnetic fields as those of a solenoid and a toroid.
A solenoid is a cylindrical coil that consists of many turns of conductor wound turn to turn on a cylindrical frame. The magnetic field of a solenoid actually consists of multiple magnetic fields of a circular current with a common axis perpendicular to the plane of each circular current.
Let's use the circulation of the magnetic induction vector and imagine the circulation along a rectangular contour 1-2-3-4 . Then the circulation of the magnetic induction vector for a given circuit will have the form
Since in the areas 2-3 And 4-1 the magnetic induction vector is perpendicular to the circuit, then the circulation is zero. Location on 3-4 , which is significantly removed from the solenoid, then it can also be ignored. Then, taking into account the law of total current, the magnetic induction in a solenoid of sufficiently large length will have the form
where n is the number of turns of the solenoid conductor per unit length,
I – current flowing through the solenoid.
A toroid is formed by winding a conductor around a ring frame. This design is equivalent to a system of many identical circular currents, the centers of which are located on a circle.
As an example, consider a toroid of radius R, on which it is wound N turns of wire. Around each turn of the wire we take a radius contour r, the center of this contour coincides with the center of the toroid. Since the magnetic induction vector B is directed tangentially to the contour at each point of the contour, then the circulation of the magnetic induction vector will have the form
where r is the radius of the magnetic induction loop.
The circuit passing inside the toroid covers N turns of wire with current I, then the law of the total current for the toroid will have the form
where n is the number of turns of the conductor per unit length,
r – radius of the magnetic induction loop,
R is the radius of the toroid.
Thus, using the law of total current and the circulation of the magnetic induction vector, it is possible to calculate an arbitrarily complex magnetic field. However, the law of total current gives correct results only in a vacuum. When calculating magnetic induction in a substance, it is necessary to take into account the so-called molecular currents. This will be discussed in the next article.
Theory is good, but without practical application it is just words.
Does the magnitude of the magnetic field induction depend on the environment in which it is formed? In order to answer this question, let's do the following experiment. Let us first determine the force (see Fig. 117) with which the magnetic field acts on a conductor with current in the air (in principle, this must be done in a vacuum), and then the force of the magnetic field on this conductor, for example, in water containing iron oxide powder ( In the figure, the vessel is shown with a dotted line). In an iron oxide medium, the magnetic field acts on the current-carrying conductor with greater force. In this case, the magnitude of the magnetic field induction is greater. There are substances, for example silver, copper, in which it is less than in a vacuum. The magnitude of the magnetic field induction depends on the environment in which it is formed.
A quantity showing how many times the magnetic field induction in a given medium is greater or less than the magnetic field induction in a vacuum is called magnetic permeability of the medium. If the induction of the magnetic field of the medium is B, and the vacuum is B 0, then the magnetic permeability of the medium
Magnetic permeability of a medium μ is a dimensionless quantity. It is different for different substances. So, for mild steel - 2180, air - 1,00000036, copper - 0,999991 . This is explained by the fact that different substances are magnetized differently in a magnetic field.
Let's find out what determines the magnetic field induction of a straight conductor carrying current. Near the straight section A of the wire turn (Fig. 122) we will place an indicator C of the magnetic field induction. Let's turn on the current. The magnetic field of section A acts on the indicator frame and rotates it, which causes the needle to deviate from the zero position. By changing the current strength in the frame with a rheostat, we notice that by how many times the current in the conductor increases, the deflection of the indicator needle increases by the same amount: V~I.
Keeping the current constant, we will increase the distance between the conductor and the frame. According to the indicator reading, we notice that the magnetic field induction is inversely proportional to the distance from the conductor to the field point being studied: V~ I/R. The magnitude of the magnetic field induction depends on the magnetic properties of the medium - on its magnetic permeability. The greater the magnetic permeability, the greater the magnetic field induction: B~μ.
Theoretically and through more accurate experiments, the French physicists Biot, Savard and Laplace established that the magnitude of the magnetic field induction of a straight wire of small cross-section in a homogeneous medium with magnetic permeability μ at a distance R from it is equal to
Here μ 0 is the magnetic constant. Let's find its numerical value and name in the SI system. Since the magnetic field induction is at the same time equal to 
then, equating these two formulas, we get
Hence the magnetic constant From the definition of ampere we know that segments of parallel conductors with a length l = 1 m while at a distance R = 1 m from each other, interact with force F = 2*10 -7 n, when current flows through them I = 1 a. Based on this, we calculate μ 0 (taking μ = 1):
Now let’s find out what determines the induction of the magnetic field inside a coil with current. Let's assemble an electrical circuit (Fig. 123). By placing the magnetic field induction indicator frame inside the coil, we close the circuit. Increasing the current strength by 2, 3 and 4 times, we notice that the magnetic field induction inside the coil increases accordingly by the same amount: V~I.
Having determined the magnetic field induction inside the coil, we increase the number of turns per unit length. To do this, connect two identical coils in series and insert one of them into the other. Using a rheostat, we will set the current strength to the previous level. With the same coil length l, the number of turns n in it has doubled and, as a consequence of this, the number of turns per unit length of the coil has doubled.
If a magnetic needle is brought close to a straight conductor carrying current, it will tend to become perpendicular to the plane passing through the axis of the conductor and the center of rotation of the needle (Fig. 67). This indicates that the needle is subject to special forces called magnetic forces. In other words, if an electric current passes through a conductor, a magnetic field appears around the conductor. A magnetic field can be considered as a special state of space surrounding current-carrying conductors.
If you pass a thick conductor through a card and pass an electric current through it, then steel filings poured onto the cardboard will be located around the conductor in concentric circles, which in this case represent the so-called magnetic lines (Fig. 68). We can move the cardboard up or down the conductor, but the location of the steel filings will not change. Consequently, a magnetic field arises around the conductor along its entire length.
If you place small magnetic arrows on the cardboard, then by changing the direction of the current in the conductor, you can see that the magnetic arrows will rotate (Fig. 69). This shows that the direction of the magnetic lines changes with the change in the direction of the current in the conductor.
The magnetic field around a current-carrying conductor has the following features: the magnetic lines of a straight conductor have the shape of concentric circles; the closer to the conductor, the denser the magnetic lines are located, the greater the magnetic induction; magnetic induction (field intensity) depends on the magnitude of the current in the conductor; The direction of the magnetic lines depends on the direction of the current in the conductor.
To show the direction of the current in the conductor shown in section, a symbol has been adopted, which we will use in the future. If you mentally place an arrow in a conductor in the direction of the current (Fig. 70), then in a conductor in which the current is directed away from us, we will see the tail of the arrow’s feathers (a cross); if the current is directed towards us, we will see the tip of an arrow (point).
The direction of magnetic lines around a current-carrying conductor can be determined by the “gimlet rule.” If a gimlet (corkscrew) with a right-hand thread moves forward in the direction of the current, then the direction of rotation of the handle will coincide with the direction of the magnetic lines around the conductor (Fig. 71).
Rice. 71. Determining the direction of magnetic lines around a current-carrying conductor using the “gimlet rule”
A magnetic needle introduced into the field of a current-carrying conductor is located along the magnetic lines. Therefore, to determine its location, you can also use the “gimlet rule” (Fig. 72).
Rice. 72. Determination of the direction of deflection of a magnetic needle brought to a conductor with current, according to the “gimlet rule”
The magnetic field is one of the most important manifestations of electric current and cannot be obtained independently and separately from the current.
In permanent magnets, the magnetic field is also caused by the movement of electrons that make up the atoms and molecules of the magnet.
The intensity of the magnetic field at each point is determined by the magnitude of magnetic induction, which is usually denoted by the letter B. Magnetic induction is a vector quantity, that is, it is characterized not only by a certain value, but also by a certain direction at each point of the magnetic field. The direction of the magnetic induction vector coincides with the tangent to the magnetic line at a given point in the field (Fig. 73).
As a result of generalizing experimental data, French scientists Biot and Savard established that magnetic induction B (magnetic field intensity) at a distance r from an infinitely long straight conductor with current is determined by the expression
where r is the radius of the circle drawn through the field point under consideration; the center of the circle is on the axis of the conductor (2πr is the circumference);
I is the amount of current flowing through the conductor.
The value μ a, which characterizes the magnetic properties of the medium, is called the absolute magnetic permeability of the medium.
For emptiness, the absolute magnetic permeability has a minimum value and is usually denoted by μ 0 and called the absolute magnetic permeability of emptiness.
1 H = 1 ohm⋅sec.
The ratio μ a / μ 0, showing how many times the absolute magnetic permeability of a given medium is greater than the absolute magnetic permeability of emptiness, is called relative magnetic permeability and is denoted by the letter μ.
The International System of Units (SI) uses the units of measurement of magnetic induction B - tesla or weber per square meter (tl, wb/m2).
In engineering practice, magnetic induction is usually measured in gauss (gs): 1 t = 10 4 gs.
If at all points of the magnetic field the magnetic induction vectors are equal in magnitude and parallel to each other, then such a field is called uniform.
The product of magnetic induction B and the area S perpendicular to the direction of the field (magnetic induction vector) is called the flux of the magnetic induction vector, or simply magnetic flux, and is denoted by the letter Φ (Fig. 74):
The International System uses the weber (wb) as the unit of measurement for magnetic flux.
In engineering calculations, magnetic flux is measured in maxwells (μs):
1 vb = 10 8 μs.
When calculating magnetic fields, a quantity called magnetic field strength (denoted H) is also used. Magnetic induction B and magnetic field strength H are related by the relation
The unit of measurement for magnetic field strength is N - ampere per meter (a/m).
The magnetic field strength in a homogeneous medium, as well as magnetic induction, depends on the magnitude of the current, the number and shape of the conductors through which the current passes. But unlike magnetic induction, magnetic field strength does not take into account the influence of the magnetic properties of the medium.
Electromagnetic phenomena
Electromagnetic phenomena reflect the connection of electric current with a magnetic field. All their physical laws are well known, and we will not try to correct them; our goal is different: to explain the physical nature of these phenomena.
One thing is already clear to us: neither electricity nor magnetism can exist without electrons; and in this electromagnetism is already manifested. We also talked about the fact that a current-carrying coil generates a magnetic field. Let's dwell on the last phenomenon and clarify how it happens.
Let's look at the coil from the end, and let the electric current flow through it counterclockwise. Current is a flow of electrons sliding along the surface of a conductor (only on the surface there are open suction grooves). The flow of electrons will carry along the adjacent ether, and it will also begin to move counterclockwise. The speed of the ether adjacent to the conductor will be determined by the speed of electrons in the conductor, and it, in turn, will depend on the difference in ether pressure (on the electrical voltage on the coil) and on the flow area of the conductor. The ether carried away by the current will affect neighboring layers, and they will also move inside and outside the coil in a circle. The speed of the swirling ether will be distributed as follows: its greatest value, of course, is in the region of the coils; when shifted towards the center, it decreases according to a linear law, so that at the very center it will be zero; When moving away from the turns to the periphery, the speed will also decrease, but not linearly, but according to a more complex law.
The macro-vortex of the ether swirled by the current will begin to orient the electrons in such a way that they will all rotate until their axes of rotation are parallel to the axis of the coil; at the same time, inside the coil they will rotate counterclockwise, and outside it - clockwise; at the same time, the electrons will tend to be coaxial, that is, they will be collected in magnetic cords. The process of electron orientation will take some time, and upon completion, a magnetic beam will appear inside the coil with the north pole in our direction, and outside the coil, on the contrary, the north pole will be far away from us. Thus, we have proven the validity of the screw or gimlet rule, known in electrical engineering, which establishes a connection between the direction of the current and the direction of the magnetic field generated by it.
The magnetic force (tension) at each point of the magnetic field is determined by the change in the speed of the ether at this point, that is, the derivative of the speed with respect to the distance from the turns of the coil: The steeper the change in speed, the greater the tension. If we correlate the magnetic force of the coil with its electrical and geometric parameters, then it has a direct dependence on the current value and an inverse dependence on the diameter of the coil. The greater the current and the smaller the diameter, the more opportunities there are to collect electrons in cords of a certain direction of rotation and the greater the magnetic force of the coil will be. It has already been said that the magnetic field strength can be enhanced or weakened by the medium.
The process of converting direct current electricity into magnetism is not reversible: if a magnet is placed in a coil, then no current arises in it. The energy of the macrovortex existing around the magnet is so small that it is unable to force electrons to move along the turns at the smallest resistance for them. Let us recall once again that in the reverse process, the macrovortex of the ether, acting as a mediator, only oriented the electrons, and nothing more, that is, it only controlled the magnetic field, and the strength of the field was determined by the number of unidirectional magnetic cords.
If you bring the magnetic needle close, it will tend to become perpendicular to the plane passing through the axis of the conductor and the center of rotation of the needle. This indicates that special forces act on the arrow, which are called magnetic forces. In addition to the effect on the magnetic needle, the magnetic field affects moving charged particles and current-carrying conductors located in the magnetic field. In conductors moving in a magnetic field, or in stationary conductors located in an alternating magnetic field, an inductive electromotive force (emf) arises.
A magnetic field
In accordance with the above, we can give the following definition of a magnetic field.
A magnetic field is one of the two sides of the electromagnetic field, excited by the electric charges of moving particles and changes in the electric field and characterized by a force effect on moving infected particles, and therefore on electric currents.
If you pass a thick conductor through cardboard and pass an electric current through it, then the steel filings poured onto the cardboard will be located around the conductor in concentric circles, which in this case are the so-called magnetic induction lines (Figure 1). We can move the cardboard up or down the conductor, but the location of the steel filings will not change. Consequently, a magnetic field arises around the conductor along its entire length.
If you put small magnetic arrows on the cardboard, then by changing the direction of the current in the conductor, you can see that the magnetic arrows will rotate (Figure 2). This shows that the direction of magnetic induction lines changes with the direction of current in the conductor.
Magnetic induction lines around a current-carrying conductor have the following properties: 1) magnetic induction lines of a straight conductor have the shape of concentric circles; 2) the closer to the conductor, the denser the magnetic induction lines are located; 3) magnetic induction (field intensity) depends on the magnitude of the current in the conductor; 4) the direction of magnetic induction lines depends on the direction of the current in the conductor.
To show the direction of the current in the conductor shown in section, a symbol has been adopted, which we will use in the future. If you mentally place an arrow in the conductor in the direction of the current (Figure 3), then in the conductor in which the current is directed away from us, we will see the tail of the arrow’s feathers (a cross); if the current is directed towards us, we will see the tip of an arrow (point).
Figure 3. Symbol for the direction of current in conductors
The gimlet rule allows you to determine the direction of magnetic induction lines around a current-carrying conductor. If a gimlet (corkscrew) with a right-hand thread moves forward in the direction of the current, then the direction of rotation of the handle will coincide with the direction of the magnetic induction lines around the conductor (Figure 4).
A magnetic needle introduced into the magnetic field of a current-carrying conductor is located along the magnetic induction lines. Therefore, to determine its location, you can also use the “gimlet rule” (Figure 5). The magnetic field is one of the most important manifestations of electric current and cannot be obtained independently and separately from the current.
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| Figure 4. Determining the direction of magnetic induction lines around a current-carrying conductor using the “gimlet rule” | Figure 5. Determining the direction of deviation of a magnetic needle brought to a conductor with current, according to the “gimlet rule” |
Magnetic induction
A magnetic field is characterized by a magnetic induction vector, which therefore has a certain magnitude and a certain direction in space.
A quantitative expression for magnetic induction as a result of generalization of experimental data was established by Biot and Savart (Figure 6). Measuring the magnetic fields of electric currents of various sizes and shapes by the deflection of the magnetic needle, both scientists came to the conclusion that every current element creates a magnetic field at some distance from itself, the magnetic induction of which is Δ B is directly proportional to the length Δ l this element, the magnitude of the flowing current I, the sine of the angle α between the direction of the current and the radius vector connecting the field point of interest to us with a given current element, and is inversely proportional to the square of the length of this radius vector r:
Where K– coefficient depending on the magnetic properties of the medium and on the chosen system of units.
In the absolute practical rationalized system of units of ICSA
where µ 0 – magnetic permeability of vacuum or magnetic constant in the MCSA system:
µ 0 = 4 × π × 10 -7 (henry/meter);
Henry (gn) – unit of inductance; 1 gn = 1 ohm × sec.
µ – relative magnetic permeability– a dimensionless coefficient showing how many times the magnetic permeability of a given material is greater than the magnetic permeability of vacuum.
The dimension of magnetic induction can be found using the formula
Volt-second is also called Weber (wb):
In practice, there is a smaller unit of magnetic induction - gauss (gs):
Biot-Savart's law allows us to calculate the magnetic induction of an infinitely long straight conductor:
Where A– the distance from the conductor to the point where the magnetic induction is determined.
Magnetic field strength
The ratio of magnetic induction to the product of magnetic permeabilities µ × µ 0 is called magnetic field strength and is designated by the letter H:
B = H × µ × µ 0 .
The last equation relates two magnetic quantities: induction and magnetic field strength.
Let's find the dimension H:
Sometimes another unit of measurement of magnetic field strength is used - Oersted (er):
1 er = 79,6 A/m ≈ 80 A/m ≈ 0,8 A/cm .
Magnetic field strength H, like magnetic induction B, is a vector quantity.
A line tangent to each point of which coincides with the direction of the magnetic induction vector is called magnetic induction line or magnetic induction line.
Magnetic flux
The product of magnetic induction by the area perpendicular to the direction of the field (magnetic induction vector) is called flux of the magnetic induction vector or simply magnetic flux and is designated by the letter F:
F = B × S .
Magnetic flux dimension:
that is, magnetic flux is measured in volt-seconds or webers.
The smaller unit of magnetic flux is Maxwell (mks):
1 wb = 108 mks.
1mks = 1 gs× 1 cm 2.
Video 1. Ampere's hypothesis
Video 1. Ampere's hypothesis
Video 2. Magnetism and electromagnetism