All elements of a circular conductor with current create magnetic fields in the center of the same direction - along the normal from the turn. therefore, all elements of the coil are perpendicular to the radius vector, then ; since the distances from all elements of the conductor to the center of the turn are the same and equal to the radius of the turn. That's why:
Direct conductor field.
As the integration constant, we choose the angle α (the angle between the vectors dB And r ), and express all other quantities through it. From the figure it follows that:
Let's substitute these expressions into the formula of the Biot-Savart-Laplace law:
And - the angles at which the ends of the conductor are visible from the point at which the magnetic induction is measured. Let's substitute it into the formula:
In the case of an infinitely long conductor ( and ) we have:
Application of Ampere's law.
Interaction of parallel currents
Consider two infinite rectilinear parallel currents directed in one direction I 1 And I 2, the distance between which is R. Each of the conductors creates a magnetic field, which acts according to Ampere's law on the other conductor with current. Current I 1 creates a magnetic field around itself, the lines of magnetic induction of which are concentric circles. Vector direction IN , is determined by the rule of the right screw, its module is equal to:
Direction of force d F 1 , with which the field B 1 acts on the area dl the second current is determined by the left-hand rule. The force modulus taking into account the fact that the angle α between the current elements I 2 and vector B 1 straight, equal
Substituting the value B 1 . we get:
By similar reasoning, one can prove that
It follows that, that is, two parallel currents are attracted to each other with the same force. If the currents are in the opposite direction, then using the left-hand rule, it can be shown that there is a repulsive force between them.
Interaction force per unit length:
Behavior of a current-carrying circuit in a magnetic field.
Let us introduce a square frame with side l with current I into the magnetic field B, the rotational moment of a pair of Ampere forces will act on the circuit:
Magnetic moment of the circuit,
Magnetic induction at the field point where the circuit is located
The current-carrying circuit tends to establish itself in a magnetic field so that the flux through it is maximum and the torque is minimum.
Magnetic induction at a given point in the field is numerically equal to the maximum torque acting at a given point in the field on a circuit with a unit magnetic moment.
Law of total current.
Let us find the circulation of vector B along a closed contour. Let's take a long conductor with current I as the field source, and a field line of radius r as the contour.
Let us extend this conclusion to a circuit of any shape, covering any number of currents. Total current law:
The circulation of the magnetic induction vector along a closed circuit is proportional to the algebraic sum of the currents covered by this circuit.
Application of the total current law to calculate fields
Field inside an infinitely long solenoid:
where τ is the linear density of winding turns, l S– solenoid length, N– number of turns.
Let the closed contour be a rectangle of length X, which braids the turns, then induction IN along this circuit:
Let's find the inductance of this solenoid:
Toroid field(wire wound around a frame in the form of a torus).
R– average radius of the torus, N– number of turns, where – linear density of winding turns.
Let's take a line of force with radius R as a contour.
Hall effect
Consider a metal plate placed in a magnetic field. An electric current is passed through the plate. A potential difference arises. Since the magnetic field acts on moving electric charges (electrons), they will be subject to the Lorentz force, moving electrons to the upper edge of the plate, and, therefore, an excess of positive charge will form at the lower edge of the plate. Thus, a potential difference is created between the upper and lower edges. The process of moving electrons will continue until the force acting from the electric field is balanced by the Lorentz force.
Where d– plate length, A– plate width, – Hall potential difference.
Law of electromagnetic induction.
Magnetic flux
where α is the angle between IN and outer perpendicular to the contour area.
For any change in magnetic flux over time. Thus, the induced emf occurs both when the area of the circuit changes and when the angle α changes. Induction emf is the first derivative of magnetic flux with respect to time:
If the circuit is closed, then an electric current begins to flow through it, called an induction current:
Where R– circuit resistance. The current arises due to a change in magnetic flux.
Lenz's rule.
An induced current always has a direction such that the magnetic flux created by this current prevents the change in the magnetic flux that caused this current. The current has such a direction as to interfere with the cause that caused it.
Rotation of the frame in a magnetic field.
Let us assume that the frame rotates in a magnetic field with an angular velocity ω, so that the angle α is equal to . in this case the magnetic flux is:
Consequently, a frame rotating in a magnetic field is a source of alternating current.
Eddy currents (Foucault currents).
Eddy currents or Foucault currents arise in the thickness of conductors that are in an alternating magnetic field, creating an alternating magnetic flux. Foucault currents lead to heating of conductors and, consequently, to electrical losses.
The phenomenon of self-induction.
With any change in magnetic flux, an induced emf occurs. Let us assume that there is an inductor through which electric current flows. According to the formula, in this case a magnetic flux is created in the coil. With any change in current in the coil, the magnetic flux changes and, therefore, an emf occurs, called self-induction emf ():
Maxwell's system of equations.
The electric field is a set of mutually related and mutually changing magnetic fields. Maxwell established a quantitative relationship between the quantities characterizing electric and magnetic fields.
Maxwell's first equation.
From Faraday's law of electromagnetic induction it follows that with any change in the magnetic flux, an emf appears. Maxwell suggested that the appearance of EMF in the surrounding space is associated with the appearance in the surrounding space vortex electromagnetic field. The conducting circuit plays the role of a device that detects the appearance of this electric field in the surrounding space.
The physical meaning of Maxwell's first equation: any change in time of the magnetic field leads to the appearance of a vortex electric field in the surrounding space.
Maxwell's second equation. Bias current.
The capacitor is connected to the DC circuit. Suppose that a circuit containing a capacitor is connected to a constant voltage source. The capacitor charges and the current in the circuit stops. If a capacitor is connected to an alternating voltage circuit, the current in the circuit does not stop. This is due to the process of continuous recharging of the capacitor, as a result of which a time-varying electric field appears between the plates of the capacitor. Maxwell suggested that a displacement current arises in the space between the plates of the capacitor, the density of which is determined by the rate of change of the electric field over time. Of all the properties inherent in electric current, Maxwell attributed one single property to displacement current: the ability to create a magnetic field in the surrounding space. Maxwell suggested that conduction current lines on the capacitor plates do not stop, but continuously transform into displacement current lines. Thus:
Thus, the current density is:
where is the conduction current density, is the displacement current density.
According to the law of total current:
The physical meaning of Maxwell's second equation: the source of the magnetic field is both conduction currents and a time-varying electric field.
Maxwell's third equation (Gauss's theorem).
The flux of the electrostatic field strength vector through a closed surface is equal to the charge contained inside this surface:
Physical meaning of Maxwell's fourth equation: lines electrostatic fields begin and end on free electric charges. That is, the source of the electrostatic field is electric charges.
Maxwell's fourth equation (magnetic flux continuity principle)
The physical meaning of Maxwell's fourth equation: the lines of the magnetic induction vector do not begin or end anywhere, they are continuous and closed on themselves.
Magnetic properties of substances.
Magnetic field strength.
The main characteristic of a magnetic field is the magnetic induction vector, which determines the force effect of the magnetic field on moving charges and currents; the magnetic induction vector depends on the properties of the medium where the magnetic field is created. Therefore, a characteristic is introduced that depends only on the currents associated with the field, but does not depend on the properties of the medium where the field exists. This characteristic is called magnetic field strength and is denoted by the letter H.
If a magnetic field in a vacuum is considered, then the intensity
where is the magnetic constant of vacuum. Unit of tension Ampere/meter.
Magnetic field in matter.
If the entire space surrounding the currents is filled with a homogeneous substance, then the magnetic field induction will change, but the distributed field will not change, that is, the magnetic field induction in the substance is proportional to the magnetic induction in vacuum. - magnetic permeability of the medium. Magnetic permeability shows how many times the magnetic field in a substance differs from the magnetic field in a vacuum. The value can be either less or greater than one, that is, the magnetic field in a substance can be either less or greater than the magnetic field in a vacuum.
Magnetization vector. Every substance is magnetic, that is, it is capable of acquiring a magnetic moment under the influence of an external magnetic field - being magnetized. The electrons of atoms under the influence of a mutual magnetic field undergo precessional motion - a movement in which the angle between the magnetic moment and the direction of the magnetic field remains constant. In this case, the magnetic moment rotates around the magnetic field with a constant angular velocity ω. Precessional motion is equivalent to circular current. Since the microcurrent is induced by an external magnetic field, then, according to Lenz’s rule, the atom has a magnetic field component directed opposite to the external field. The induced component of magnetic fields adds up and forms its own magnetic field in the substance, directed opposite to the external magnetic field, and, therefore, weakening this field. This effect is called the diamagnetic effect, and substances in which the diamagnetic effect occurs are called diamagnetic substances or diamagnetic substances. In the absence of an external magnetic field, a diamagnetic material is nonmagnetic, since the magnetic moments of the electrons are mutually compensated and the total magnetic moment of the atom is zero. Since the diamagnetic effect is caused by the action of an external magnetic field on the electrons of the atoms of a substance, diamagnetism is characteristic of ALL SUBSTANCES.
Paramagnetic substances are substances in which, even in the absence of an external magnetic field, atoms and molecules have their own magnetic moment. However, in the absence of an external magnetic field, the magnetic moments of different atoms and molecules are randomly oriented. In this case, the magnetic moment of any macroscopic volume of matter is zero. When a paramagnetic substance is introduced into an external magnetic field, the magnetic moments are oriented in the direction of the external magnetic field, and a magnetic moment appears directed along the direction of the magnetic field. However, the total magnetic field arising in a paramagnetic substance significantly overlaps the diamagnetic effect.
The magnetization of a substance is the magnetic moment per unit volume of the substance.
where is the magnetic moment of the entire magnet, equal to the vector sum of the magnetic moments of individual atoms and molecules.
The magnetic field in a substance consists of two fields: an external field and a field created by the magnetized substance:
(reads "hee") is the magnetic susceptibility of the substance.
Let's substitute formulas (2), (3), (4) into formula (1):
The coefficient is a dimensionless quantity.
For diamagnetic materials (this means that the field of molecular currents is opposite to the external field).
For paramagnetic materials (this means that the field of molecular currents coincides with the external field).
Therefore, for diamagnetic materials, and for paramagnetic materials. And N .
Hysteresis loop.
Magnetization dependence J on the strength of the external magnetic field H forms a so-called “hysteresis loop”. At the beginning (section 0-1) the ferromagnet is magnetized, and the magnetization does not occur linearly, and at point 1 saturation is achieved, that is, with a further increase in the magnetic field strength, the current growth stops. If you start to increase the strength of the magnetizing field, then the decrease in magnetization follows the curve 1-2 , lying above the curve 0-1 . When residual magnetization is observed (). The existence of permanent magnets is associated with the presence of residual magnetization. The magnetization goes to zero at point 3, at a negative value of the magnetic field, which is called the coercive force. With a further increase in the opposite field, the ferromagnet is remagnetized (curve 3-4). Then the ferromagnet can be demagnetized again (curve 4-5-6) and magnetize again until saturation (curve 6-1). Ferromagnets with low coercivity (with small values of ) are called soft ferromagnets, and they correspond to a narrow hysteresis loop. Ferromagnets with a high coercive force are called hard ferromagnets. For each ferromagnet there is a certain temperature, called the Curie point, at which the ferromagnet loses its ferromagnetic properties.
The nature of ferromagnetism.
According to Weiss's ideas. Ferromagnets at temperatures below the Curie point have a domain structure, namely, ferromagnets consist of macroscopic regions called domains, each of which has its own magnetic moment, which is the sum of the magnetic moments of a large number of atoms of a substance oriented in the same direction. In the absence of an external magnetic field, the domains are randomly oriented and the resulting magnetic moment of the ferromagnet is generally zero. When an external magnetic field is applied, the magnetic moments of the domains begin to be oriented in the direction of the field. In this case, the magnetization of the substance increases. At a certain value of the external magnetic field strength, all domains are oriented along the field direction. In this case, the growth of magnetization stops. When the external magnetic field strength decreases, the magnetization begins to decrease again; however, not all domains are misoriented at the same time, so the decrease in magnetization occurs more slowly, and when the magnetic field strength is equal to zero, a fairly strong orienting connection remains between some domains, which leads to the presence of residual magnetization coinciding with direction of the previously existing magnetic field.
To break this connection, it is necessary to apply a magnetic field in the opposite direction. At temperatures above the Curie point, the intensity of thermal motion increases. Chaotic thermal movement breaks the bonds within the domains, that is, the preferential orientation of the domains themselves is lost. Thus, the ferromagnet loses its ferromagnetic properties.
Exam questions:
1) Electric charge. Law of conservation of electric charge. Coulomb's law.
2) Electric field strength. The physical meaning of tension. Field strength of a point charge. Electric field lines.
3) Two definitions of potentials. Work on moving a charge in an electric field. The connection between tension and potential. Work along a closed trajectory. Circulation theorem.
4) Electrical capacity. Capacitors. Series and parallel connection of capacitors. Capacitance of a parallel plate capacitor.
5) Electric current. Conditions for the existence of electric current. Current strength, current density. Units of current measurement.
6) Ohm's law for a homogeneous section of the chain. Electrical resistance. Dependence of resistance on the cross-sectional length of the conductor material. Dependence of resistance on temperature. Serial and parallel connection of conductors.
7) Outside forces. EMF. Potential difference and voltage. Ohm's law for a non-uniform section of a circuit. Ohm's law for a closed circuit.
8) Heating of conductors with electric current. Joule-Lenz law. Electric current power.
9) Magnetic field. Ampere power. Left hand rule.
10) Movement of a charged particle in a magnetic field. Lorentz force.
11) Magnetic flux. Faraday's law of electromagnetic induction. Lenz's rule. The phenomenon of self-induction. Self-induced emf.
Magnetic field of current:
A magnetic field created around electric charges as they move. Since the movement of electric charges represents an electric current, around any conductor with current there is always current magnetic field.
To verify the existence of a magnetic field of current, let’s bring an ordinary compass from above to the conductor through which electric current flows. The compass needle will immediately deviate to the side. We bring the compass to the conductor with current from below - the compass needle will deviate in the other direction (Figure 1).
Let us apply the Biot–Savart–Laplace law to calculate the magnetic fields of the simplest currents. Let's consider the magnetic field of direct current.
All vectors dB from arbitrary elementary sections dl have the same direction. Therefore, addition of vectors can be replaced by addition of modules.
Let the point at which the magnetic field is determined be located at a distance b from the wire. From the figure it can be seen that:
;
Substituting the found values r and d l into the Biot-Savart-Laplace law, we get:
For final conductor angle α varies from , to. Then
For infinitely long conductor , and , then
or, which is more convenient for calculations, .
Direct current magnetic induction lines are a system of concentric circles enclosing the current.
21. Biot-Savart-Laplace law and its application to the calculation of the magnetic field induction of a circular current.
Magnetic field of a circular conductor carrying current.
22. Magnetic moment of a coil with current. Vortex nature of the magnetic field.
The magnetic moment of a coil with current is a physical quantity, like any other magnetic moment, that characterizes the magnetic properties of a given system. In our case, the system is represented by a circular coil with current. This current creates a magnetic field that interacts with the external magnetic field. This can be either the field of the earth or the field of a permanent or electromagnet.
Figure - 1 circular turn with current
A circular coil with current can be represented as a short magnet. Moreover, this magnet will be directed perpendicular to the plane of the coil. The location of the poles of such a magnet is determined using the gimlet rule. According to which the north plus will be located behind the plane of the coil if the current in it moves clockwise.
Figure-2 Imaginary strip magnet on the coil axis
This magnet, that is, our circular coil with current, like any other magnet, will be affected by an external magnetic field. If this field is uniform, then a torque will arise that will tend to turn the coil. The field will rotate the coil so that its axis is located along the field. In this case, the field lines of the coil itself, like a small magnet, must coincide in direction with the external field.
If the external field is not uniform, then translational motion will be added to the torque. This movement will occur due to the fact that sections of the field with higher induction will attract our magnet in the form of a coil more than areas with lower induction. And the coil will begin to move towards the field with greater induction.
The magnitude of the magnetic moment of a circular coil with current can be determined by the formula.
Where, I is the current flowing through the turn
S area of the turn with current
n normal to the plane in which the coil is located
Thus, from the formula it is clear that the magnetic moment of a coil is a vector quantity. That is, in addition to the magnitude of the force, that is, its modulus, it also has a direction. The magnetic moment received this property due to the fact that it includes the normal vector to the plane of the coil.
dl
RdB,B
It is easy to understand that all current elements create a magnetic field of the same direction in the center of the circular current. Since all elements of the conductor are perpendicular to the radius vector, due to which sinα = 1, and are located at the same distance from the center R, then from equation 3.3.6 we obtain the following expression
B = μ 0 μI/2R. (3.3.7)
2. Direct current magnetic field infinite length. Let the current flow from top to bottom. Let us select several elements with current on it and find their contributions to the total magnetic induction at a point located at a distance from the conductor R. Each element will give its own vector dB , directed perpendicular to the plane of the sheet “towards us”, the total vector will also be in the same direction IN . When moving from one element to another, which are located at different heights of the conductor, the angle will change α ranging from 0 to π. Integration will give the following equation
B = (μ 0 μ/4π)2I/R. (3.3.8)
As we said, the magnetic field orients the current-carrying frame in a certain way. This happens because the field exerts a force on each element of the frame. And since the currents on opposite sides of the frame, parallel to its axis, flow in opposite directions, the forces acting on them turn out to be in different directions, as a result of which a torque arises. Ampere established that the force dF , which acts from the field side on the conductor element dl , is directly proportional to the current strength I in the conductor and the cross product of an element of length dl for magnetic induction IN :
dF = I[dl , B ]. (3.3.9)
Expression 3.3.9 is called Ampere's law. The direction of the force vector, which is called Ampere force, are determined by the rule of the left hand: if the palm of the hand is positioned so that the vector enters it IN , and direct the four extended fingers along the current in the conductor, then the bent thumb will indicate the direction of the force vector. Ampere force modulus is calculated by the formula
dF = IBdlsinα, (3.3.10)
Where α – angle between vectors d l And B .
Using Ampere's law, you can determine the strength of interaction between two currents. Let's imagine two infinite straight currents I 1 And I 2, flowing perpendicular to the plane of Fig. 3.3.4 towards the observer, the distance between them is R. It is clear that each conductor creates a magnetic field in the space around itself, which, according to Ampere’s law, acts on another conductor located in this field. Let's select on the second conductor with current I 2 element d l and calculate the force d F 1 , with which the magnetic field of a current-carrying conductor I 1 affects this element. Lines of magnetic induction field that creates a current-carrying conductor I 1, are concentric circles (Fig. 3.3.4).
IN 1
d F 2d F 1
B 2
Vector IN 1 lies in the plane of the figure and is directed upward (this is determined by the rule of the right screw), and its modulus
B 1 = (μ 0 μ/4π)2I 1 /R. (3.3.11)
Force d F 1 , with which the field of the first current acts on the element of the second current, is determined by the left-hand rule, it is directed towards the first current. Since the angle between the current element I 2 and vector IN 1 direct, for the modulus of force taking into account 3.3.11 we obtain
dF 1= I 2 B 1 dl= (μ 0 μ/4π)2I 1 I 2 dl/R. (3.3.12)
It is easy to show, by similar reasoning, that the force dF 2, with which the magnetic field of the second current acts on the same element of the first current
Let a direct electric current of force I flow along a flat circular contour of radius R. Let us find the field induction in the center of the ring at point O
Let us mentally divide the ring into small sections that can be considered rectilinear, and apply the Biot-Savarre-Laplace law to determine the induction of the field created by this element in the center of the ring. In this case, the vector of the current element (IΔl)k and the vector rk connecting this element with the observation point (the center of the ring) are perpendicular, therefore sinα = 1. The induction vector of the field created by the selected section of the ring is directed along the axis of the ring, and its modulus is equal to
For any other element of the ring, the situation is absolutely similar - the induction vector is also directed along the axis of the ring, and its module is determined by formula (1). Therefore, the summation of these vectors is carried out elementary and is reduced to the summation of the lengths of the sections of the ring 
Let's complicate the problem - find the field induction at point A, located on the axis of the ring at a distance z from its center. 
As before, we select a small section of the ring (IΔl)k and construct the induction vector of the field ΔBk created by this element at the point in question. This vector is perpendicular to the vector r connecting the selected area with the observation point. Vectors (IΔl)k and rk, as before, are perpendicular, so sinα = 1. Since the ring has axial symmetry, the total field induction vector at point A must be directed along the axis of the ring. The same conclusion about the direction of the total induction vector can be reached if we notice that each selected section of the ring has a symmetrical one on the opposite side, and the sum of two symmetrical vectors is directed along the axis of the ring. Thus, in order to determine the module of the total induction vector, it is necessary to sum up the projections of the vectors onto the axis of the ring. This operation is not particularly difficult, given that the distances from all points of the ring to the observation point are the same rk = √(R2+ z2), and the angles φ between the vectors ΔBk and the axis of the ring are the same. Let us write down the expression for the modulus of the desired total induction vector 
From the figure it follows that cosφ = R/r, taking into account the expression for the distance r, we obtain the final expression for the field induction vector 
As one would expect, in the center of the ring (at z = 0) formula (3) transforms into the previously obtained formula (2).
Using the general method discussed here, it is possible to calculate the field induction at an arbitrary point. The system under consideration has axial symmetry, so it is enough to find the field distribution in a plane perpendicular to the plane of the ring and passing through its center. Let the ring lie in the xOy plane (Fig. 433), 
and the field is calculated in the yOz plane. The ring should be divided into small sections visible from the center at an angle Δφ and the fields created by these sections should be summed up. It can be shown (try it yourself) that the components of the magnetic induction vector of the field created by one selected current element at a point with coordinates (y, z) are calculated using the formulas:
Let us consider the expression for the field induction on the ring axis at distances significantly larger than the ring radius z >> R. In this case, formula (3) is simplified and takes the form
Where IπR2 = IS = pm is the product of the current strength and the area of the circuit, that is, the magnetic moment of the ring. This formula coincides (if, as usual, replace μo in the numerator with εo in the denominator) with the expression for the electric field strength of a dipole on its axis.
This coincidence is not accidental; moreover, it can be shown that such a correspondence is valid for any point in the field located at large distances from the ring. In fact, a small circuit with current is a magnetic dipole (two identical small oppositely directed current elements) - therefore its field coincides with the field of an electric dipole. To more clearly emphasize this fact, a picture of the magnetic field lines of the ring at large distances from it is shown (compare with a similar picture for the field of an electric dipole).
Magnetic field strength on the axis of a circular current (Fig. 6.17-1) created by a conductor element IDl, is equal
because in this case
Rice. 6.17. Magnetic field on the circular current axis (left) and electric field on the dipole axis (right)
When integrated over a turn, the vector will describe a cone, so that as a result only the field component along the axis will “survive” 0z. Therefore, it is enough to sum up the value
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Integration
is carried out taking into account the fact that the integrand does not depend on the variable l, A
Accordingly, complete magnetic induction on the coil axis equal to
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In particular, in the center of the turn ( h= 0) field is equal
At a great distance from the coil ( h >> R) we can neglect the unit under the radical in the denominator. As a result we get
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Here we have used the expression for the magnitude of the magnetic moment of a turn Р m, equal to the product I per area of the turn. The magnetic field forms a right-handed system with the circular current, so (6.13) can be written in vector form
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For comparison, let's calculate the field of an electric dipole (Fig. 6.17-2). The electric fields from positive and negative charges are equal, respectively,
so the resulting field will be
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At long distances ( h >> l) we have from here
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Here we used the concept of the vector of the electric moment of a dipole introduced in (3.5). Field E parallel to the dipole moment vector, so (6.16) can be written in vector form
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The analogy with (6.14) is obvious.
Power lines circular magnetic field with current are shown in Fig. 6.18. and 6.19
Rice. 6.18. Magnetic field lines of a circular coil with current at short distances from the wire
Rice. 6.19. Distribution of magnetic field lines of a circular coil with current in the plane of its symmetry axis.
The magnetic moment of the coil is directed along this axis
In Fig. 6.20 presents an experiment in studying the distribution of magnetic field lines around a circular coil with current. A thick copper conductor is passed through holes in a transparent plate on which iron filings are poured. After turning on a direct current of 25 A and tapping on the plate, the sawdust forms chains that repeat the shape of the magnetic field lines.
The magnetic lines of force for a coil whose axis lies in the plane of the plate are concentrated inside the coil. Near the wires they have a ring shape, and far from the coil the field quickly decreases, so that the sawdust is practically not oriented.
Rice. 6.20. Visualization of magnetic field lines around a circular coil with current
Example 1. An electron in a hydrogen atom moves around a proton in a circle of radius a B= 53 pm (this value is called the Bohr radius after one of the creators of quantum mechanics, who was the first to calculate the orbital radius theoretically) (Fig. 6.21). Find the strength of the equivalent circular current and magnetic induction IN fields in the center of the circle.
Rice. 6.21. Electron in a hydrogen atom and B = 2.18·10 6 m/s. A moving charge creates a magnetic field at the center of the orbit
The same result can be obtained using expression (6.12) for the field at the center of the coil with a current, the strength of which we found above
Example 2. An infinitely long thin conductor with a current of 50 A has a ring-shaped loop with a radius of 10 cm (Fig. 6.22). Find the magnetic induction at the center of the loop.
Rice. 6.22. Magnetic field of a long conductor with a circular loop
Solution. The magnetic field at the center of the loop is created by an infinitely long straight wire and a ring coil. The field from a straight wire is directed orthogonally to the plane of the drawing “at us”, its value is equal to (see (6.9))
The field created by the ring-shaped part of the conductor has the same direction and is equal to (see 6.12)
The total field at the center of the coil will be equal to
Additional Information
http://n-t.ru/nl/fz/bohr.htm - Niels Bohr (1885–1962);
http://www.gumer.info/bibliotek_Buks/Science/broil/06.php - Bohr's theory of the hydrogen atom in Louis de Broglie's book “Revolution in Physics”;
http://nobelprize.org/nobel_prizes/physics/laureates/1922/bohr-bio.html - Nobel Prizes. Nobel Prize in Physics 1922 Niels Bohr.