We cannot see the magnetic field, but for a better understanding of magnetic phenomena, it is important to learn how to depict it. Magnetic arrows will help with this. Each such arrow is a small permanent magnet that easily rotates in a horizontal plane (Fig. 2.1). About how the magnetic field is graphically depicted and what physical quantity characterizes it, you will learn from this paragraph.
Rice. 2.2. In a magnetic field, magnetic needles are oriented in a certain way: North Pole the arrows indicate the direction of the magnetic field induction vector at a given point
We study the power characteristic of the magnetic field
If a charged particle moves in a magnetic field, then the field will act on the particle with some force. The value of this force depends on the charge of the particle, the direction and value of the speed of its movement, and also on how strong the field is.
The power characteristic of a magnetic field is magnetic induction.
Magnetic induction (magnetic field induction) is a vector physical quantity that characterizes the force action of a magnetic field.
Magnetic induction is denoted by the symbol B.
The SI unit of magnetic induction is tesla; named after the Serbian physicist Nikola Tesla (1856-1943):
The direction of the magnetic induction vector at a given point of the magnetic field is taken to be the direction indicated by the north pole of the magnetic needle installed at this point (Fig. 2.2).
Note! The direction of the force with which the magnetic field acts on moving charged particles or on a conductor with current, or on a magnetic needle, does not coincide with the direction of the magnetic induction vector.
Magnetic lines:
Rice. 2.3. Magnetic field lines of a bar magnet
Outside the magnet, they leave the north pole of the magnet and enter the south;
Always closed (the magnetic field is a vortex field);
Most densely located at the poles of the magnet;
Never cross
Depicting a magnetic field
On fig. 2.2 we see how the magnetic needles are oriented in a magnetic field: their axes seem to form lines, and the magnetic induction vector at each point is directed along the tangent to the line passing through this point.
By using magnetic lines graphically depict magnetic fields:
1) the direction of the magnetic induction vector is taken as the direction of the magnetic induction line at a given point;
Rice. 2.4. Chains of iron filings reproduce the pattern of magnetic induction lines of the magnetic field of a horseshoe magnet
2) the larger the magnetic induction module, the closer the magnetic lines are drawn to each other.
Having considered the graphical representation of the magnetic field of a bar magnet, we can draw some conclusions (see Fig. 2.3).
Note that these conclusions are valid for the magnetic lines of any magnet.
What is the direction of the magnetic lines inside the bar magnet?
The picture of magnetic lines can be reproduced using iron filings.
Let's take a horseshoe-shaped magnet, put a Plexiglas plate on it and pour iron filings onto the plate through a strainer. In a magnetic field, each piece of iron will become magnetized and turn into a small "magnetic needle". Improvised "arrows" will be oriented along the magnetic lines of the magnetic field of the magnet (Fig. 2.4).
Draw a picture of the magnetic lines of the magnetic field of a horseshoe magnet.
Learn about a uniform magnetic field
A magnetic field in some part of space is called homogeneous if at each of its points the magnetic induction vectors are the same both in absolute value and in direction (Fig. 2.5).
In areas where the magnetic field is uniform, the lines of magnetic induction are parallel and located on the same distance from each other (Fig. 2.5, 2.6). It is customary to depict the magnetic lines of a uniform magnetic field directed towards us as dots (Fig. 2.7, a) - as if we see “arrowheads” flying towards us. If the magnetic lines are directed away from us, then they are depicted as crosses - we seem to see the “feathers of arrows” flying from us (Fig. 2.7, b).
In most cases, we are dealing with an inhomogeneous magnetic field, a field in which different points which the magnetic induction vectors have different meanings and directions. The magnetic lines of such a field are curved, and their density is different.
Rice. 2.6. The magnetic field inside the bar magnet (a) and between two magnets facing each other with opposite poles (b) can be considered homogeneous
Studying the Earth's magnetic field
To study terrestrial magnetism, William Gilbert made a permanent magnet in the form of a ball (a model of the Earth). Having placed a compass on the ball, he noticed that the compass needle behaves in the same way as on the surface of the Earth.
The experiments allowed the scientist to assume that the Earth is a huge magnet, and its south magnetic pole is located in the north of our planet. Further research confirmed W. Gilbert's hypothesis.
On fig. 2.8 shows a picture of the lines of magnetic induction of the Earth's magnetic field.
rice. 2.7. The image of the lines of magnetic induction of a uniform magnetic field, which are perpendicular to the plane of the figure and directed towards us (a); sent from us (b)
Imagine that you are walking towards the North Pole, moving exactly in the direction that the compass needle is pointing. Will you reach your destination?
The lines of magnetic induction of the Earth's magnetic field are not parallel to its surface. If you fix the magnetic needle in the gimbal suspension, that is, so that it can rotate freely both around the horizontal and
Rice. 2.8. The layout of the magnetic lines of the magnetic field of the planet Earth
and around the vertical axes, the arrow will be set at an angle to the Earth's surface (Fig. 2.9).
How will the magnetic needle be located in the device in fig. 2.9 near the earth's north magnetic pole? near the earth's south magnetic pole?
The Earth's magnetic field has long helped to orient travelers, sailors, the military and not only them. It is proved that fish, marine mammals and birds during their migrations are guided by the Earth's magnetic field. They also orient themselves, looking for a way home, and some animals, such as cats.
Learn about magnetic storms
Studies have shown that in any area the Earth's magnetic field periodically, every day, changes. In addition, small annual changes in the Earth's magnetic field are observed. However, there are also drastic changes. Strong disturbances of the Earth's magnetic field, which cover the entire planet and last from one to several days, are called magnetic storms. Healthy people practically do not feel them, but those who have cardiovascular diseases and diseases nervous system, magnetic storms cause deterioration in well-being.
The Earth's magnetic field is a kind of "shield" that protects our planet from charged particles flying from space, mainly from the Sun ("solar wind"). Near the magnetic poles, particle streams fly quite close to the Earth's atmosphere. With an increase in solar activity, cosmic particles enter the upper layers of the atmosphere and ionize gas molecules - auroras are observed on Earth (Fig. 2.10).
Summing up
Magnetic induction B is a vector physical quantity that characterizes the force action of a magnetic field. The direction of the magnetic induction vector coincides with the direction indicated by the north pole of the magnetic needle. The SI unit of magnetic induction is tesla (T).
Conditional directed lines, at each point of which the tangent coincides with the line along which the magnetic induction vector is directed, are called lines of magnetic induction or magnetic lines.
The lines of magnetic induction are always closed, outside the magnet they leave the north pole of the magnet and enter the south, they are denser in those areas of the magnetic field where the magnetic induction module is greater.
Planet Earth has a magnetic field. Near the geographic north pole of the Earth is its south magnetic pole, near the geographic south pole - the north magnetic pole.
Control questions
1. Define magnetic induction. 2. How is the magnetic induction vector directed? 3. What is the SI unit of magnetic induction? Who is she named after? 4. Give the definition of lines of magnetic induction. 5. What direction is taken as the direction of the magnetic lines? 6. What determines the density of magnetic lines? 7. What magnetic field is called homogeneous? 8. Prove that the Earth has a magnetic field. 9. How are the Earth's magnetic poles relative to the geographic ones? 10. What are magnetic storms? How do they affect a person?
Exercise #2
1. In fig. 1 shows the lines of magnetic induction in a certain section of the magnetic field. For each cases a-c determine: 1) whether this field is homogeneous or non-homogeneous; 2) the direction of the magnetic induction vector at points A and B of the field; 3) at which point - A or B - the magnetic induction of the field is greater.
2. Why can a steel window grill become magnetized over time?
3. In fig. 2 shows the lines of the magnetic field created by two identical permanent magnets facing each other with the same poles.
1) Is there a magnetic field at point A?
2) What is the direction of the magnetic induction vector at point B? at point C?
3) At what point - A, B or C - is the magnetic field induction the greatest?
4) What is the direction of the magnetic induction vectors inside the magnets?
4. Previously, during expeditions to the North Pole, there were difficulties in determining the direction of movement, because ordinary compasses almost did not work near the pole. Why do you think?
5. Use additional sources of information and find out how important the magnetic field is for life on our planet. What would happen if the Earth's magnetic field suddenly disappeared?
6. There are areas of the earth's surface where the magnetic induction of the earth's magnetic field is much greater than in neighboring areas. Use additional sources of information and learn more about magnetic anomalies.
7. Explain why any uncharged body is always attracted to a body that has an electric charge.
This is textbook material.
« Physics - Grade 11 "
The electric field is characterized by intensity electric field.
The electric field strength is a vector quantity. The magnetic field is characterized by magnetic induction.
Magnetic induction is a vector quantity, it is denoted by the letter.
Direction of the magnetic induction vector
The direction of the magnetic induction vector is the direction that shows the north pole N of the magnetic needle, which is freely installed in the magnetic field.
This direction coincides with the direction of the positive normal to the closed loop with current.
Using a frame with a current or a magnetic needle, you can determine the direction of the magnetic induction vector at any point in the field.
In the magnetic field of a rectilinear conductor with current, a magnetic needle at each point is set tangentially to a circle whose plane is perpendicular to the wire, and its center lies on the axis of the wire.
gimlet rule
The direction of the magnetic induction vector is set using the gimlet rule.
If the direction of the translational movement of the gimlet coincides with the direction of the current in the conductor, then the direction of rotation of the gimlet handle indicates the direction of the magnetic induction vector
Lines of magnetic induction
The magnetic field can be shown using lines of magnetic induction.
Lines of magnetic induction are called lines, the tangents to which at any point coincide with the vector at a given point of the field. The lines of the vector of magnetic induction are similar to the lines of the vector of the electrostatic field.
Lines of magnetic induction can be made visible using iron filings.
Magnetic field of a straight conductor with current
For a straight conductor with current, the lines of magnetic induction are concentric circles lying in a plane perpendicular to this conductor with current. The center of the circles is on the axis of the conductor. The arrows on the lines indicate in which direction the magnetic induction vector is directed, tangent to this line.
Magnetic field of current coil (solenoid)
If the length of the solenoid is much greater than its diameter, then the magnetic field inside the solenoid can be considered homogeneous.
Lines of magnetic induction of such a field are parallel and are at equal distances from each other.
Earth's magnetic field
The lines of magnetic induction of the Earth's field are similar to the lines of magnetic induction of the field of a solenoid.
The Earth's magnetic axis makes an angle of 11.5° with the Earth's axis of rotation.
Periodically, the magnetic poles change their polarity.
Vortex field
The lines of force of an electrostatic field always have sources: they start on positive charges and end on negative ones.
And the lines of magnetic induction have neither beginning nor end, they are always closed.
Fields with closed vector lines are called eddy.
The magnetic field is a vortex field.
The magnetic field has no sources.
Magnetic charges, similar to electric ones, do not exist in nature.
So, the magnetic field is a vortex field, at each point the magnetic induction vector is indicated by a magnetic arrow, the direction of the magnetic induction vector can be determined by the gimlet rule
A line drawn in a magnetic field so that at any point the tangent coincides with the induction vector ( and Fig. 119, a) of the magnetic field at this point is called magnetic field induction line. To get a picture of the lines of induction, it is necessary to place a large number of magnetic needles in a magnetic field. The location of the arrows will show the shape of the lines of induction. Iron filings are taken as such arrows, which are magnetized in a magnetic field and, interacting with each other, interlock with their ends, forming chains depicting induction lines. The direction of the induction line is taken to be the direction that shows the north pole of the magnetic needle at a given location in the field. Therefore, the induction vector at a given point of the field has a direction coinciding with the direction of the induction line drawn through this point.
The lines of induction of a direct conductor with current represent concentric circles located in planes perpendicular to the direction of the current, and the centers of all these circles are on the axis of the conductor (see Fig. 118, b). Their direction is determined by the gimlet rule. A direct current magnetic field has no magnetic poles. The lines of induction, the magnetic field of a coil with current inside it are parallel (see Fig. 119, b), but outside the coil they are not parallel. A coil with current has two magnetic poles. Its polarity, and therefore the direction of the induction lines inside the coil, is determined by the rule of grabbing it with your right hand (Fig. 119, c): if you take the coil with your right hand so that four fingers indicate the direction of the current, then the thumb along the coil will point to the end of the coil, which is the north magnetic pole, and will also show the direction of the induction lines inside the coil. The magnetic fields of a coil with current and a permanent magnet are identical. Northern and South Pole s exist only in pairs - it is impossible to get one pole.
As in the case of an electrostatic field, only one line of induction can be drawn through each point in space. Therefore, these lines do not cross each other anywhere. In contrast to the lines of the electrostatic field (see Fig. 50), the lines of magnetic field induction are closed lines of both the magnetic field of the current and the permanent magnet (Fig. 119, d). The closedness of the induction lines indicates that the magnetic field is vortex. They always cover the current or moving charge with which the magnetic field is associated. Some of the induction lines are closed in the immediate vicinity of the current, others are far from it, and then it seems to us that they go to infinity at both ends (see Fig. 119, b, d).
We agreed to draw the lines of induction so that the number of lines passing through the area unit, perpendicular to the induction vector at a given point, was equal to the value of the field induction at this place. Magnetic spectra give an idea of the distribution of magnetic induction in magnitude and direction.
Based on the induction formula, we set the unit of measurement of the magnetic field induction in international system units:
The unit of magnetic field induction tesla is the induction of such a homogeneous magnetic field in which a straight conductor 1 m long, with a current of 1 a, located perpendicular to the induction lines *, is acted upon by a force of 1 n (Fig. 120, a). On fig. 120b shows the magnetometer measuring the magnitude of the magnetic field of a permanent magnet.
* (Under this condition, the force will be maximum.)
The induction of the Earth's magnetic field is small: near the equator, about 32*10 -6 tl, at the poles - 65*10 -6 tl, in the region of the Kursk magnetic anomaly - 190*10 -6 tl. At present, in laboratories with the help of coils, magnetic fields with induction up to 15 tl.
Does the magnitude of the induction of the magnetic field of the current depend on the shape of the conductor? Between the sides of a conductor shaped as in fig. 121, a, place a magnetic needle and connect the conductor to a current source. We observe a large deviation of the arrow. Having made the conductor straight (Fig. 121, b) and placing a magnetic needle under it, let us pass a current through it, as in the first case. Note a slight deviation of the arrow. We twist the conductor, as shown in Fig. 121, in; we see that the arrow does not deviate, that is, the twisted (bifilar) conductor does not have a magnetic field. The greater the induction of the magnetic field, the stronger it acts on the magnetic needle. From the experiments we conclude: the magnitude of the induction of the magnetic field of the current depends on the shape of the conductor: a\u003e b, c \u003d 0. Ceteris paribus, the magnitude of the magnetic field induction is greatest for a conductor in the form of a coil.
Magnetic field - a component of the electromagnetic field that appears in the presence of a time-varying electric field. In addition, a magnetic field can be created by the current of charged particles, or by the magnetic moments of electrons in atoms (permanent magnets).
Magnetic induction-vector quantity, which is the force characteristic of the magnetic field at a given point in space. Shows the force with which the magnetic field acts on a charge moving with speed.
Lines of magnetic induction(field lines of a magnetic field) are called lines drawn in a magnetic field so that at each point of the field the tangent to the line of magnetic induction coincides with the direction of the vector IN at this point in the field.
The lines of magnetic induction are easiest to observe with the help of small
Needle-shaped iron filings that are magnetized in the field under study and behave like small magnetic needles (a free magnetic needle turns in a magnetic field so that the axis of the arrow connecting its south pole to north coincides with the direction IN).
The view of the lines of magnetic induction of the simplest magnetic fields is shown
in fig. From fig. b- G it can be seen that these lines encircle a current-carrying conductor that creates a field. Near the conductor, they lie in planes perpendicular to the conductor.
H 
the direction of the induction lines is determined by gimlet rule: if you screw the gimlet in the direction of the current density vector in the conductor, then the direction of movement of the gimlet handle will indicate the direction of the lines of magnetic induction.
Lines of magnetic field induction
the current at any point cannot be interrupted, i.e., neither begin nor end: they are either closed (Fig. b, c, d) or they wind endlessly around some surface, densely filling it everywhere, but never returning a second time to any point of the surface.
Gauss' theorem for magnetic induction
Flux of the magnetic induction vector through any closed surface zero:
This is equivalent to the fact that in nature there are no "magnetic charges" (monopoles) that would create a magnetic field, as electric charges create an electric field. In other words, the Gauss theorem for magnetic induction shows that the magnetic field is eddy.
2 Bio-Savart-Laplace law
Let the direct current flow along the contour γ, which is in vacuum, - the point at which the field is sought, then the magnetic field induction at this point is expressed by the integral (in the SI system)
The direction is perpendicular, that is, perpendicular to the plane in which they lie, and coincides with the tangent to the line of magnetic induction. This direction can be found by the rule of finding the lines of magnetic induction (right screw rule): the direction of rotation of the screw head gives the direction if the translational movement of the gimlet corresponds to the direction of the current in the element. The module of the vector is determined by the expression (in the SI system)
The vector potential is given by the integral (in the SI system)
The Biot-Savart-Laplace law can be obtained from the Maxwell equations for a stationary field. In this case, the time derivatives are equal to 0, so that the equations for the field in vacuum take the form (in the CGS system)
where is the current density in space. In this case, the electric and magnetic fields are independent. Let's use the vector potential for the magnetic field (in the CGS system):
The gauge invariance of the equations makes it possible to impose one additional condition on the vector potential:
Expanding the double rotor using the vector analysis formula, we obtain an equation of the Poisson type for the vector potential:
Its particular solution is given by an integral similar to the Newtonian potential:
Then the magnetic field is determined by the integral (in the CGS system)
similar in form to the Biot-Savart-Laplace law. This correspondence can be made exact if we use generalized functions and write down the spatial current density corresponding to the loop with current in empty space. Passing from integration over the entire space to the iterated integral along the loop and along its orthogonal planes and taking into account that
we obtain the Biot - Savart - Laplace law for the field of a coil with current.
For a visual representation of the magnetic field, magnetic induction lines are used. Line of magnetic induction they call such a line, at each point of which the magnetic field induction (vector) is directed tangentially to the curve. The direction of these lines coincides with the direction of the field. It was agreed to draw the lines of magnetic induction so that the number of these lines per unit area of the area perpendicular to them would be equal to the modulus of induction in the given region of the field. Then the magnetic field is judged by the density of the lines of magnetic induction. Where the lines are thicker, the modulus of the magnetic field is greater. Lines of magnetic induction are always closed Unlike lines of electrostatic field strength, which are open (begin and end on charges). The direction of the lines of magnetic induction is found according to the rule of the right screw: if the translational motion of the screw coincides with the direction of the current, then its rotation occurs in the direction of the lines of magnetic induction. As an example, let's give a picture of the lines of magnetic induction of a direct current flowing perpendicular to the plane of the drawing from us beyond the drawing (Fig. 2).
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Rice. 3 Let us find the circulation of the magnetic field induction along a circle of arbitrary radius a, coinciding with the line of magnetic induction. The field is created by the current force I, flowing along an infinitely long conductor located perpendicular to the plane of the drawing (Fig. 3). The magnetic field induction is directed tangentially to the magnetic induction line. We transform the expression, since a = 0 and cosa = 1. The induction of the magnetic field created by the current flowing through an infinitely long conductor is calculated by the formula: B= m0m I/(2p a), That 
The circulation of the vector along this contour, we find by the formula (3): 
m 0 m I, because 
- circumference. So, 
It can be shown that this relation is valid for an arbitrary-shaped contour enclosing a current-carrying conductor. If the magnetic field is created by a system of currents I 1, I 2, ... , I n, then the circulation of the magnetic field induction along a closed circuit covering these currents is equal to
(4)
Relation (4) is the law of total current: the circulation of the magnetic field induction along an arbitrary closed circuit is equal to the product of the magnetic constant, magnetic permeability and algebraic sum the strength of the currents covered by this circuit.
The current strength can be found using the current density j: Where S is the cross-sectional area of the conductor. Then the total current law is written as
(5)
MAGNETIC FLOW.
By analogy with the flux of the electric field strength, the magnetic field induction flux or magnetic flux is introduced. Magnetic flux through some surface call the number of lines of magnetic induction penetrating it. Let there be a surface with area S. For finding magnetic flux through it we mentally divide the surface into elementary sections with an area dS, which can be considered flat, and the field within them is uniform (Fig. 4). Then the elementary magnetic flux dФ Through this surface is equal to: dФ B = B dS cos a = B n dS, Where B- modulus of magnetic field induction at the site location, a - angle between the vector and the normal to the site, B n = B cos a is the projection of the magnetic field induction onto the direction of the normal. magnetic flux F B through the entire surface is equal to the sum of these flows dФ B, i.e.
  Rice. 4 |
(6)
because the summation of infinitesimals is an integration.
In SI units, magnetic flux is measured in webers (Wb). 1 Wb \u003d 1 T 1 m 2.
THE GAUSS THEOREM FOR A MAGNETIC FIELD
In electrodynamics, the following theorem is proved: the magnetic flux penetrating an arbitrary closed surface is zero , i.e.
This ratio is called Gauss theorems for the magnetic field. This theorem is a consequence of the fact that in nature there are no "magnetic charges" (unlike electric ones) and the lines of magnetic induction are always closed (unlike the lines of the electrostatic field strength, which begin and end on electric charges).
WORK ON MOVING A CONDUCTOR WITH CURRENT IN A MAGNETIC FIELD
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It is known that the Ampère force acts on a current-carrying conductor in a magnetic field. If the conductor moves, then when it moves, this force does work. Let's define it for a particular case. Consider an electrical circuit, one of the sections DC which can slide (without friction) over the contacts. In this case, the chain forms a flat contour. This circuit is in a uniform magnetic field with an induction perpendicular to the plane of the circuit, directed at us (Fig. 5). To the plot DC Ampere's force works
F = BIl sina =BIl, (8)
Where l- section length, I- the current flowing through the conductor. - the angle between the directions of the current and the magnetic field. (In this case, a = 90° and sin a = 1). The direction of the force is found by the rule of the left hand. When moving an area DC at an elementary distance dx elementary work is done dA equal to dA = F dx. Considering (8), we get:
dA = BIl dx = IB dS = I dF B, (9)
because the dS = l dx- the area described by the conductor during its movement, dФ B =B dS- magnetic flux through this area or change in magnetic flux through the area of a flat closed loop. Expression (9) is also valid for an inhomogeneous magnetic field. Thus, work to move a closed loop with direct current in a magnetic field is equal to the product of the current strength and the change in the magnetic flux through the area of this circuit.
THE PHENOMENON OF ELECTROMAGNETIC INDUCTION
The phenomenon of electromagnetic induction is as follows: with any change in the magnetic flux penetrating the area covered by the conducting circuit, a electromotive force . They call her emf induction . If the circuit is closed, then under the action of emf. appears electricity, named induction .
Consider one of the experiments carried out by Faraday to detect the induction current, and therefore the emf. induction. If a magnet is inserted or extended into a solenoid closed to a very sensitive electrical measuring device (galvanometer) (Fig. 6), then when the magnet moves, a deflection of the galvanometer needle is observed, indicating the occurrence of an induction current. The same is observed when the solenoid moves relative to the magnet. If the magnet and the solenoid are stationary relative to each other, then the induction current does not occur. Thus, with the mutual motion of these bodies, the magnetic flux created by the magnetic field of the magnet changes through the turns of the solenoid, which leads to the appearance of an induction current caused by the emerging emf. induction.
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LENTZ RULE
The direction of the induction current is determined Lenz's rule :inductive current always has such a direction that the magnetic field it creates prevents the change in magnetic flux that causes this current. It follows from this that with an increase in the magnetic flux, the resulting inductive current will have such a direction that the magnetic field generated by it is directed against the external field, counteracting the increase in the magnetic flux. A decrease in the magnetic flux, on the contrary, leads to the appearance of an induction current that creates a magnetic field that coincides in direction with the external field.
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Let, for example, in a uniform magnetic field there is a square frame made of metal and penetrated by a magnetic field (Fig. 7). Let's assume that the magnetic field increases. This leads to an increase in the magnetic flux through the frame area. According to Lenz's rule, the magnetic field of the resulting inductive current will be directed against the external field, i.e. the vector of this field is opposite to the vector . Applying the rule of the right screw (if the screw is rotated so that its translational movement coincides with the direction of the magnetic field, then its rotary motion gives the direction of the current), we find the direction of the induction current II.
THE LAW OF ELECTROMAGNETIC INDUCTION.
The law of electromagnetic induction, which determines the emerging emf, was discovered by Faraday empirically. However, it can be obtained based on the law of conservation of energy.
Back to electrical circuit shown in fig. 5 placed in a magnetic field. Let's find the work done by the current source with emf. e for an elementary period of time dt, when moving charges along the chain. From the definition of emf. Job dA external forces is equal to: dA stor = e dq, Where dq- the amount of charge flowing through the circuit during the time dt. But dq = I dt, Where I- current in the circuit. Then
dA stor = e I dt. (10)
The work of the current source is spent on the release of a certain amount of heat dQ and to work dA according to the movement of the conductor DC in a magnetic field. According to the law of conservation of energy, the equality
dA stor = dQ + dA.(11)
From the Joule-Lenz law we write:
dQ = I 2R dt, (12)
Where R is the total resistance of this circuit, and from expression (9)
dA = I dФ B, (13)
Where dФ B is the change in the magnetic flux through the area of a closed loop when the conductor moves. Substituting expressions (10), (12) and (13) into formula (12), after reducing by I, we get e· dt = IR dt + dФ B. Dividing both sides of this equality by dt, we find: I = (e- From this expression it follows that in the circuit, in addition to emf. e, there is some other electromotive force ei equal to
(14)
and due to a change in the magnetic flux penetrating the circuit area. This emf and is the emf. electromagnetic induction or short emf. induction. Relation (14) is law of electromagnetic induction, which is formulated: emf induction in the circuit is equal to the rate of change of the magnetic flux penetrating the area covered by this circuit. The minus sign in formula (14) is a mathematical expression of the Lenz rule.
