If conductors with currents of the same direction are located close to one another, then the magnetic lines of these conductors, covering both conductors, having the property of longitudinal tension and tending to shorten, will force the conductors to attract (Fig. 90, a).

Magnetic lines two conductors with currents of different directions in the space between the conductors are directed in the same direction. Magnetic lines that have the same direction will repel each other. Therefore, conductors with currents of the opposite direction repel each other (Fig. 90, b).

Consider the interaction of two parallel conductors with currents located at a distance a from one another. Let the length of the conductors be l.

The magnetic induction created by the current I 1 on the location line of the second conductor is equal to

An electromagnetic force will act on the second conductor

The magnetic induction created by the current I 2 on the location line of the first conductor will be equal to

and an electromagnetic force acts on the first conductor

equal in magnitude to the force F 2

Force acting from the side magnetic field on the charges moving in it, are called Lorentz force.

The Lorentz force is determined by the relation:

F l \u003d q V B sina

where q is the value of the moving charge;
V is the module of its speed;
B is the modulus of the magnetic field induction vector;
a is the angle between the charge velocity vector and the magnetic induction vector.

Please note that the Lorentz force is perpendicular to the velocity and therefore it does not do work, does not change the modulus of the charge velocity and its kinetic energy. But the direction of the speed changes continuously

The Lorentz force is perpendicular to the vectors B and v, and its direction is determined using the same left-hand rule as the direction of the Ampère force: if left hand arrange so that the component of magnetic induction B, perpendicular to the charge velocity, enters the palm, and four fingers are directed along the movement positive charge(against the movement of the negative), then the thumb bent 90 degrees will show the direction of the Lorentz force acting on the charge F l.

The Lorentz force depends on the moduli of particle velocity and magnetic field induction. This force is perpendicular to the velocity and therefore determines the centripetal acceleration of the particle. The particle moves uniformly along a circle of radius r

Hall effect- the phenomenon of the occurrence of a transverse potential difference (also called the Hall voltage) when a conductor with direct current is placed in a magnetic field. Discovered by Edwin Hall in 1879 in thin gold plates.

In its simplest form, the Hall effect looks like this. Let flow through a metal bar in a weak magnetic field B electricity under the action of tension E. The magnetic field will deflect charge carriers (for definiteness, electrons) from their movement along or against electric field to one of the edges of the beam. In this case, the criterion of smallness will be the condition that in this case the electron does not begin to move along the cycloid.

Thus, the Lorentz force will lead to the accumulation negative charge near one face of the bar and positive near the opposite. The accumulation of charge will continue until the resulting electric field of charges E1 compensates for the magnetic component of the Lorentz force:

The electron velocity v can be expressed in terms of the current density:

where n is the concentration of charge carriers. Then

The coefficient of proportionality between E1 and jB is called the Hall coefficient (or constant). In this approximation, the sign of the Hall constant depends on the sign of the charge carriers, which makes it possible to determine their type for a large number metals. For some metals (for example, such as aluminum, zinc, iron, cobalt), in strong fields, a positive sign of RH is observed, which is explained in the semiclassical and quantum theories solid body.

Electromagnetic induction- the phenomenon of the occurrence of an electric current in a closed circuit when the magnetic flux through it changes.

Electromagnetic induction was discovered by Michael Faraday in 1831. He discovered that electromotive force, arising in a closed conducting circuit is proportional to the rate of change of the magnetic flux through the surface bounded by this circuit. The value of emf does not depend on what causes the change in the flux - a change in the magnetic field itself or the movement of a circuit (or part of it) in a magnetic field. The electric current caused by this emf. , is called the induced current.

Faraday's law

According to the law electromagnetic induction Faraday:

Where is the electromotive force acting along an arbitrarily chosen contour,

Magnetic flux through a surface stretched over this contour.

The minus sign in the formula reflects Lenz's rule,

Lenz's rule, a rule for determining the direction induction current: Induction current, arising from the relative motion of the conducting circuit and the source of the magnetic field, always has such a direction that its own magnetic flux compensates for changes in the external magnetic flux that caused this current. Formulated in 1833 by E. Kh. Lenz.

If the current increases, then the magnetic flux increases.

If a the induction current is directed against the main current.

If the induction current is directed in the same direction as the main current.

The induction current is always directed in such a way as to reduce the effect of the cause that causes it.

For a coil in an alternating magnetic field, Faraday's law can be written as follows:

Where is the electromotive force,

Number of turns

Magnetic flux through one turn,

Coil flux linkage.

vector shape

In differential form, Faraday's law can be written as follows:

self induction- occurrence phenomenon EMF induction in a conducting circuit when the current flowing through the circuit changes.

When the current in the circuit changes, the magnetic flux through the surface bounded by this circuit changes, the change in the flux of magnetic induction leads to the excitation of the EMF of self-induction. The direction of the EMF turns out to be such that when the current in the circuit increases, the EMF prevents the increase in current, and when the current decreases, it prevents it from decreasing.

The value of the EMF is proportional to the rate of change of the current strength I and the inductance of the circuit L:

Due to the phenomenon of self-induction in electrical circuit with an EMF source, when the circuit is closed, the current is not established instantly, but after some time. Similar processes occur when the circuit is opened, while the value of the self-induction emf can significantly exceed the source emf.

Solenoid Inductance

A solenoid is a long, thin coil, that is, a coil whose length is much greater than its diameter. Under these conditions and without the use of magnetic material, the magnetic flux density B inside the coil is virtually constant and equal to

where μ0 is the vacuum permeability, N is the number of turns, i is the current, and l is the length of the coil. Neglecting the edge effects at the ends of the solenoid, we find that the flux linkage through the coil is equal to the flux density B multiplied by the cross-sectional area S and the number of turns N:

From here follows the formula for the inductance of the solenoid

Magnetic field energies

The increment in the energy density of the magnetic field is:

In an isotropic linear magnet:

where: μ - relative magnetic permeability

In vacuum μ = 1 and:

The energy of the magnetic field in the inductor can be found by the formula:

Φ - magnetic flux,

L is the inductance of a coil or coil with current.

Displacement currents

To describe and explain the "passage" of alternating current through a capacitor (discontinuity in direct current) Maxwell introduced the concept of displacement current.

The displacement current also exists in the conductors through which flows alternating current conduction, but in this case it is negligible compared to the conduction current. The presence of displacement currents was confirmed experimentally by the Soviet physicist A. A. Eikhenvald, who studied the magnetic field of the polarization current, which is part of the displacement current. In the general case, the conduction currents and displacements in space are not separated, they are in the same volume. Therefore, Maxwell introduced the concept of total current, equal to the sum conduction currents (as well as convection currents) and displacement. Total current density:

To distinguish between the conduction current and the displacement current, it is customary to designate different symbols - i and j.

In a dielectric (for example, in a dielectric of a capacitor) and in a vacuum, there are no conduction currents. Therefore, Maxwell's equation is written as -

This restores the historical validity of Maxwell, when he determined that light is an electromagnetic wave with vectors H and E -

Maxwell's equations- a system of differential equations describing the electromagnetic field and its relationship with electric charges and currents in vacuum, and continuous media. Together with the expression for the Lorentz force, they form complete system equations of classical electrodynamics. The equations formulated by James Clerk Maxwell on the basis of the experimental results accumulated by the middle of the 19th century played an important role in the emergence of the special theory of relativity.

Differential form

Gauss law

Electric charge is the source of electrical induction. Gauss' law for a magnetic field

There are no magnetic charges.[~ 1] Faraday's law of induction

The change in magnetic induction generates a vortex electric field. [~ 1] Ampere - Maxwell's law

Electric current and change in electric induction generate a vortex magnetic field

integral form

Using the Ostrogradsky-Gauss and Stokes formulas differential equations Maxwell can be given the form of integral equations:

Gauss law

The flow of electrical induction through a closed surface s is proportional to the amount of free charge in the volume v that surrounds the surface s. Gauss' law for a magnetic field

The flux of magnetic induction through a closed surface is zero (magnetic charges do not exist). Faraday's law of induction

The change in the flux of magnetic induction passing through the open surface s, taken with the opposite sign, is proportional to the circulation of the electric field on the closed contour l, which is the boundary of the surface s. Ampère-Maxwell law

The total electric current of free charges and the change in the flow of electric induction through the open surface s are proportional to the circulation of the magnetic field on the closed contour l, which is the boundary of the surface s.

 - a two-dimensional closed surface in the case of the Gauss theorem, limiting the volume, and an open surface in the case of the Faraday and Ampère-Maxwell laws (its boundary is a closed contour).

 - electric charge enclosed in a volume bounded by a surface

 - electric current passing through the surface

Material Equations

Material equations establish a connection between and . In this case, the individual properties of the environment are taken into account. In practice, the material equations usually use experimentally determined coefficients (depending in the general case on the frequency electromagnetic field), which are collected in various reference books physical quantities

Border conditions are obtained from Maxwell's equations by passing to the limit. To do this, the easiest way is to use Maxwell's equations in integral form.

Choosing in the second pair of equations the integration contour in the form of a rectangular frame of infinitely small height crossing the interface between two media, we can obtain the following relationship between the field components in two regions adjacent to the boundary

where is the unit normal to the surface directed from medium 1 to medium 2, is the density of surface currents along the boundary. The first boundary condition can be interpreted as continuity at the boundary of the regions of the tangential components of the electric field strengths (it follows from the second that the tangential components of the magnetic field strength are continuous only in the absence of surface currents at the boundary).

Similarly, choosing the domain of integration in the first pair of integral equations in the form of a cylinder of infinitely small height crossing the interface so that its generators are perpendicular to the interface, one can obtain:

de is the surface charge density.

These boundary conditions show the continuity of the normal component of the magnetic induction vector (the normal component of the electric induction is continuous only if there are no surface charges on the boundary).

From the continuity equation, one can obtain the boundary condition for currents:

,

An important special case is the interface between a dielectric and an ideal conductor. Since an ideal conductor has infinite conductivity, the electric field inside it is zero (otherwise it would generate an infinite current density). Then, in the general case of variable fields, it follows from Maxwell's equations that the magnetic field in the conductor is zero. As a result, the tangential component of the electric and normal magnetic fields at the boundary with ideal conductor are equal to zero:

sinusoidal current, alternating current, which is a sinusoidal function of time of the form: i \u003d Im sin (wt + j), where i is the instantaneous value of the current, Im is its amplitude, w is the angular frequency, j is the initial phase. Since the sinusoidal function has a similar derivative, then in all parts of the linear circuit Sinusoidal current voltages, currents and induced emfs are also sinusoidal. Appropriateness of application Sinusoidal current in technology is associated with the simplification of electrical devices and circuits (as well as their calculations).

Independent work

A magnetic field

1 option

2. An electron flies into a magnetic field with an induction of 1.4 * 10-3 T in vacuum at a speed of 500 km / s perpendicular to the lines of magnetic induction. Determine the force acting on the electron and the radius of the circle along which it moves.

Independent work

A magnetic field

Option 2

2.

Independent work

A magnetic field

1 option

1. What force acts on a conductor 0.1 m long in a uniform magnetic field with a magnetic induction of 2 T, if the current in the conductor is 5 A, and the angle between the direction of the current and the lines of induction is 300?

3. Determine the magnitude and direction of the Lorentz force acting on the proton in the case shown in the figure. B = 80 mT, υ = 200 km/h.

4. Is it possible to transport hot steel ingots in the workshop of a metallurgical plant using an electromagnet?

5. A proton accelerated in an electric field by a potential difference of 1.5 * 105 V flies into a uniform magnetic field perpendicular to the lines of magnetic induction and moves uniformly along a circle with a radius of 0.6 m. Determine the proton speed, the magnitude of the magnetic induction vector and the force with which it acts to a proton.

Independent work

A magnetic field

3 option

1. Calculate the induction of the magnetic field in which a maximum force of 10 mN acts on a conductor 0.3 m long at a current of 0.5 A.

2. In a uniform magnetic field with an induction of 1 T, a proton moves at a speed of 108 m/s perpendicular to the lines of induction. Determine the force acting on the proton and the radius of the circle along which it moves.

3. Determine the strength and direction of the current in the case shown in the figure. B = 50 mT, FA = 40 mN.

4. Why two parallel wires, through which currents pass in opposite directions, repel each other?

5. Accelerated in an electric field with a potential difference of 4.5 * 103 V, an electron flies into a uniform magnetic field and moves along a helix with a radius of 30 cm and a step of 8 cm. Determine the magnetic field induction.

Independent work

A magnetic field

1 option

1. What force acts on a conductor 0.1 m long in a uniform magnetic field with a magnetic induction of 2 T, if the current in the conductor is 5 A, and the angle between the direction of the current and the lines of induction is 300?

2. An electron flies into a magnetic field with an induction of 1.4 * 10-3 T in vacuum at a speed of 500 km / s perpendicular to the lines of magnetic induction. Determine the force acting on the electron and the radius of the circle along which it moves.

3. Determine the magnitude and direction of the Lorentz force acting on the proton in the case shown in the figure. B = 80 mT, υ = 200 km/h.

4. Is it possible to transport hot steel ingots in the workshop of a metallurgical plant using an electromagnet?

5. A proton accelerated in an electric field by a potential difference of 1.5 * 105 V flies into a uniform magnetic field perpendicular to the lines of magnetic induction and moves uniformly along a circle with a radius of 0.6 m. Determine the proton speed, the magnitude of the magnetic induction vector and the force with which it acts to a proton.

Independent work

A magnetic field

Option 2

1. Calculate the Lorentz force acting on a proton moving at a speed of 106 m/s in a uniform magnetic field with an induction of 0.3 T perpendicular to the lines of induction.

2. In a uniform magnetic field with an induction of 0.8 T, a conductor with a current of 30 A, the length of the active part of which is 10 cm, is acted upon by a force of 1.5 N. At what angle to the magnetic induction vector is the conductor placed?

3. Determine the magnitude and direction of the magnetic induction vector in the case shown in the figure. Υ = 10 Mm/s, FL = 0.5 pN.

4. Why doesn't the Lorentz force do work?

5. A charged particle moves in a magnetic field with an induction of 3 T along a circle with a radius of 4 cm at a speed of 106 m/s. Find the particle charge if its energy is 12000 eV.

We already know, firstly, that a current-carrying conductor creates a magnetic field around itself, and secondly, that a current-carrying conductor, being in a magnetic field, is subjected to a force.

The following consequence follows from this: two wires with current must act on one another. Indeed, consider two parallel wires, the currents in which have opposite directions (Fig. 2.16).

The current of the first of them creates a magnetic field around itself, shown in Fig. 2.36 with one circular line. This line goes through the second wire. By applying the left hand rule to the second wire, it is easy to see that it repels from the first.

The force with which the first current, directed towards us, acts on the second, is equal in magnitude and opposite in direction to the force with which the second current acts on the first.

Rice. 2.16. Wires with oppositely directed currents repel each other. The figure shows the cross section of the wires by the plane of the drawing. The direction of the currents is schematically depicted by a dot (the tip of the arrow directed towards us) and a cross (the tail of the arrow directed away from us). The ring line shows the magnetic field of the first current. Applying the left hand rule to determine the force acting on the second wire, you need to place your left hand palm down and extend four fingers towards the drawing. A bent thumb will show that the force is directed to the right.

there are repulsive forces between currents directed in opposite directions. There are attractive forces between currents of the same direction.

We leave the proof of this to the reader (Fig. 2.17).

Rice. 2.17. Wires with the same direction of currents attract each other

Calculation of the force of interaction of rectilinear parallel wires.

Let us show how the force of interaction of two rectilinear parallel wires flowed around by currents is calculated. Around a straight wire with current I a magnetic field is created, the induction of which is equal to

Here d is the distance from the axis of the wire to that point of the field in which we are looking for induction; ; it is obvious that the greater this distance, the lower the corresponding value of the magnetic induction.

By measuring the current in amperes, and the distance d in meters, we obtain the value of the magnetic induction in teslas.

If in the magnetic field created by the current I, there is another wire with a current G, then the force acting on it is equal to (see formula § 2.5)

Let us calculate the force of interaction between two wires, the distance between which is 20 cm, under short circuit conditions, i.e. at a very high current, for example 30,000 A. The first wire creates a field whose induction at a distance of 20 cm turns out to be

If the length of the wires is 1 m, then the force of interaction of the wires