Let's apply Ampere's law to calculate the force of interaction between two long straight conductors with currents I 1 and I 2 located at a distance d from each other (Fig. 6.26).
Rice. 6.26. Power interaction of rectilinear currents:
1 - parallel currents; 2 - antiparallel currents
Conductor with current I 1 creates a ring magnetic field, the magnitude of which at the location of the second conductor is equal to
This field is directed “away from us” orthogonally to the plane of the drawing. The element of the second conductor experiences the action of the Ampere force from the side of this field
Substituting (6.23) into (6.24), we get
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With parallel currents the strength F 21 is directed towards the first conductor (attraction), with antiparallel - in reverse side(repulsion).
Similarly, conductor element 1 is affected by the magnetic field created by the current-carrying conductor I 2 at a point in space with an element with force F 12. Reasoning in the same way, we find that F 12 = –F 21, that is, in this case Newton’s third law is satisfied.
So, the interaction force of two straight infinitely long parallel conductors, calculated per element of the length of the conductor, is proportional to the product of the current forces I 1 and I 2 flowing in these conductors, and is inversely proportional to the distance between them. In electrostatics, two long charged threads interact according to a similar law.
In Fig. Figure 6.27 presents an experiment demonstrating the attraction of parallel currents and the repulsion of antiparallel ones. For this purpose, two aluminum strips are used, suspended vertically next to each other in a slightly tensioned state. When parallel direct currents of about 10 A are passed through them, the ribbons are attracted. and when the direction of one of the currents changes to the opposite, they repel.
Rice. 6.27. Force interaction of long straight conductors with current
Based on formula (6.25), the unit of current is established - ampere, which is one of the basic units in SI.
Example. Along two thin wires, bent in the form of identical rings with a radius R= 10 cm, equal currents flow I= 10 A each. The planes of the rings are parallel, and the centers lie on a straight line orthogonal to them. The distance between centers is d= 1 mm. Find the forces of interaction between the rings.
Solution. In this problem it should not be confusing that we only know the law of interaction of long straight conductors. Since the distance between the rings is much less than their radius, the interacting elements of the rings “do not notice” their curvature. Therefore, the interaction force is given by expression (6.25), where we must substitute the circumference of the rings. We then obtain
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 contract, 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 having the same direction will repel each other. Therefore, conductors with currents of opposite directions repel one another (Fig. 90, b).
Let us 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 line of location of the second conductor is equal to
The second conductor will be subject to an electromagnetic force
The magnetic induction created by the current I 2 on the line of location of the first conductor will be equal to
and the first conductor is acted upon by an electromagnetic force
equal in magnitude to force F2
The operating principle of electrodynamic measuring instruments is based on the electromechanical interaction of conductors with current; used in direct and especially alternating current circuits.
Problems to solve independently
1. Determine tension magnetic field, created by a current of 100 A, passing along a long straight conductor at a point distant from the conductor by 10 cm.
2. Determine the strength of the magnetic field created by the current 20 A, passing along a ring conductor with a radius of 5 cm at a point located in the center of the coil.
3. Define magnetic flux, passing through a piece of nickel placed in a uniform magnetic field of strength 500 car The cross-sectional area of a piece of nickel is 25 ohm 2 (relative permeability of nickel 300).
4. Straight conductor length 40 cm placed in a uniform magnetic field at an angle of 30°C to the direction of the magnetic field. Passes along the conductor § current 50 A. The field induction is 5000 ee. Determine the force with which a conductor is pushed out of a magnetic field.
5. Determine the force with which two straight, parallel conductors located in the air repel each other. Conductor length 2 m, the distance between them is 20 cm. Currents in conductors 10 each A.
Security questions
1. How can you verify that a magnetic field is formed around a current-carrying conductor?
2. What are the properties of magnetic lines?
3. How to determine the direction of magnetic lines?
4. What is a solenoid called and what is its magnetic field?
5. How to determine the poles of a solenoid?
6. What is an electromagnet called and how to determine its poles?
7. What is hysteresis?
8. What are the shapes of electromagnets?
9. How do conductors through which electric current flows interact with each other?
10.What acts on a current-carrying conductor in a magnetic field?
11.How to determine the direction of the force acting on a current-carrying conductor in a magnetic field?
12.What principle is the operation of electric motors based on?
13.What bodies are called ferromagnetic?
The Biot-Savart-Laplace and Ampere laws are used to determine the force of interaction of two parallel conductors with current. Consider two infinite straight conductors with currents I1 and I2, the distance between which is equal to a. In Fig. 1.10 the conductors are located perpendicular to the drawing. The currents in them are directed equally (due to the drawing on us) and are indicated by dots. Each conductor creates a magnetic field that acts on the other conductor. Current I1 creates a magnetic field around itself, the lines of magnetic induction of which are concentric circles. Direction 
is determined by the right-hand screw rule, and its modulus is determined by the Biot-Savart-Laplace law. According to the above calculations, the module is equal to 
Then, according to Ampere's law, dF1=I2B1dl or 
and similarly 
. N 
direction of force 
, with which the field 
acts on section dℓ of the second conductor with current I 2 (Fig. 1.10), determined by the left-hand rule (see Section 1.2). As can be seen from Fig. 1.10 and calculations, the forces 
identical in magnitude and opposite in direction. In our case, they are directed towards each other and the conductors attract. If currents flow in opposite directions, then the forces arising between them repel the conductors from each other. So, parallel currents (one direction) attract, and antiparallel currents (opposite directions) repel. To determine the force F acting on a conductor of finite length ℓ, it is necessary to integrate the resulting equality over ℓ from 0 to ℓ: 
At magnetic interaction the law of action and reaction is fulfilled, i.e. Newton's third law:
.
1.5. The effect of a magnetic field on a moving charged particle.@
As already noted, the most important feature of a magnetic field is that it acts only on moving electric charges. As a result of experiments, it was established that any charged particle moving in a magnetic field experiences a force F, which is proportional to the magnitude of the magnetic field at this point. The direction of this force is always perpendicular to the speed of the particle and depends on the angle between the directions 
. This force is called Lorentz force. The modulus of this force is equal to 
where q is the amount of charge; v – speed of its movement; 
– vector of magnetic field induction; α – angle between vectors 
And 
. IN vector form the expression for the Lorentz force has the form 
.
For the case when the charge speed is perpendicular to the magnetic induction vector, the direction of this force is determined using the left hand rule: if the palm of the left hand is positioned so that the vector 
entered the palm, and directed the fingers along 
(for q>0), then the thumb bent at a right angle will indicate the direction of the Lorentz force for q>0 (Fig. 1.11, a). For q< 0 сила Лоренца имеет противоположное
направление (рис.1.11,б).
Since this force is always perpendicular to the speed of the particle, it changes only the direction of the speed, and not its magnitude, and therefore the Lorentz force does not do any work. That is, the magnetic field does not do work on a charged particle moving in it and its kinetic energy does not change during such a movement.
The particle deflection caused by the Lorentz force depends on the sign of q. This is the basis for determining the sign of the charge of particles moving in magnetic fields. A magnetic field does not act on a charged particle ( 
) in two cases: if the particle is motionless ( 
) or if the particle moves along the magnetic field line. In this case, the vectors 
parallel and sinα=0. If the velocity vector 
perpendicular 
, then the Lorentz force creates centripetal acceleration and the particle will move in a circle. If the speed is directed at an angle to 
, then the charged particle moves in a spiral, the axis of which is parallel to the magnetic field.
The work of all charged particle accelerators is based on this phenomenon - devices in which beams of high-energy particles are created and accelerated under the influence of electric and magnetic fields.
The action of the Earth's magnetic field near the Earth's surface changes the trajectory of particles emitted by the Sun and stars. This explains the so-called latitude effect, which consists in the fact that the intensity of cosmic rays reaching the Earth near the equator is less than at higher latitudes. The action of the Earth's magnetic field explains the fact that the aurora is observed only at the highest latitudes, in the Far North. It is in this direction that the Earth's magnetic field deflects charged cosmic particles, which cause the glow in the atmosphere called the aurora.
In addition to the magnetic force, the already familiar electric force can also act on the charge. 
, and the resulting electromagnetic force acting on the charge has the form
E 
that formula is called Lorentz formula. For example, electrons in cathode ray tubes of televisions, radars, electronic oscilloscopes, and electron microscopes are exposed to this force.
The law of total current for a magnetic field in vacuum.
Vector circulation theorem or total current law for magnetic field in vacuum is formulated as follows: the circulation of a vector along an arbitrary closed loop is equal to the product of the magnetic constant by algebraic sum currents covered by this circuit, i.e.
Where n is the number of conductors with currents covered by a circuit l of arbitrary shape.
Magnetic field of a toroid and salenoid.
Magnetic field on the direct axis long solenoid.
Solenoid is a coil wound on a cylindrical frame. If length solenoid much more its diameter, then such a solenoid is called long(unlike short coil with opposite size ratios). Magnetic field maximum inside the solenoid and directed along its axis. Near the axis of the solenoid, the magnetic field can be considered homogeneous. To find the magnetic field strength on the axis of a straight long solenoid using the magnetic field circulation theorem, we select an integration contour, as shown in Fig. 10.5.
Fig.10.5.
In section 1-2, the direction of the magnetic field coincides with the direction of traversing the circuit, and its strength is constant due to the uniformity of the field. In sections 2-3 and 4-1 outside the solenoid, the projection of the magnetic field onto the bypass direction is zero. Finally, in section 3-4, which is far enough away from the solenoid, we can assume that there is no magnetic field.
Taking into account the above we have:
But according to the theorem about magnetic voltage this integral is equal to , where N– the number of solenoid turns coupled to the integration circuit. Hence
where we find: ,
where denotes the number of turns per unit length of the solenoid.
Calculation of magnetic induction of an infinitely long solenoid:
2)Magnetic field on the toroid axis.
Toroid is a coil wound on a torus-shaped frame. The magnetic field of the toroid is entirely concentrated inside it and is heterogeneous. Maximum value The magnetic field strength is on the axis of the toroid.
Fig.10.6. Towards the calculation of the magnetic field strength on the toroid axis.
To find the magnetic field strength near the toroid axis, we apply the magnetic field circulation theorem by choosing an integration contour, as shown in Fig. 10.6.
.
On the other hand, this integral is equal to , which means that
Calculation of magnetic induction of a toroid:
Ampere's law
The force with which a magnetic field acts on an element of a current-carrying conductor located in a magnetic field is directly proportional to the strength of the current I in a conductor and the vector product of the conductor length element and magnetic induction:
The direction of the force is determined by the calculation rule vector product, which is convenient to remember using the left-hand rule.
The ampere force modulus can be found using the formula:
where α is the angle between the magnetic induction and current vectors.
Strength dF maximum when the conductor element with current is located perpendicular to the lines of magnetic induction ():
Two infinite parallel conductors in a vacuum
Most famous example The following problem illustrates the Ampere force. In a vacuum at a distance r two infinite parallel conductors are located from each other, in which currents flow in one direction I 1 and I 2. It is required to find the force acting per unit length of the conductor.
Infinite Explorer with current I 1 at a point in the distance r creates a magnetic field with induction:
(according to the Biot-Savart-Laplace law).
Now, using Ampere’s law, we find the force with which the first conductor acts on the second:
According to the gimlet rule, it is directed towards the first conductor (similarly for, which means the conductors attract).
Modulus of a given force ( r- distance between conductors):
We integrate, taking into account only a conductor of unit length (limits l from 0 to 1).
