Laws of electromagnetism. Laws of electromagnetism - Miracles of ordinary things

Translation of an article fromhttp://www.coilgun.eclipse.co.uk/ by Roman.

Fundamentals of electromagnetism

In this section, we will look at general electromagnetic principles that are widely used in engineering. This is a very brief introduction to such a complex topic. You must find yourself a good book on magnetism and electromagnetism if you want to understand this section better. You can also find most of these concepts detailed in Fizzics Fizzle (http://library.thinkquest.org/16600/advanced/electricityandmagnetism.shtml).

electromagnetic fieldsandstrength

Before we consider a special case - coilgun -a, we need to briefly familiarize ourselves with the basics of electromagnetic fields and forces. Whenever there is a moving charge, there is a corresponding magnetic field associated with it. It can arise due to the current in the conductor, the rotation of an electron in its orbit, the plasma flow, etc. To facilitate the understanding of electromagnetism, we use the concept of the electromagnetic field and magnetic poles. Differential vector equations that describe this field have been developed James Clark Maxwell.

1. Measurement systems

Just to make life more difficult, there are three measurement systems that are popularly used. They're called Sommerfield, Kennely and Gaussian . Since each system has different elements (names) for many of the same things, it can be confusing. I will use Sommerfield The system shown below:

Quantity
Field (Tension)
magnetic flux		weber (W)
Induction		tesla(T)
Magnetization
Magnetization intensity
Moment

Table 1 Measurement system

2. LawBio- Savara

Using the Biot-Savart law, you can determine the magnetic field created by an elementary current .

Figure 2.1

Ex.. 2.1

where H field component at a distance r , created by current i , current in the elementary section of the conductor of length l . u unit vector directed radially from l .

We can determine the magnetic field created by the combination of several elementary currents using this law. Consider an infinitely long conductor carrying a current i . We can use Biot-Savart's law to obtain a basic solution for the field at any distance from the conductor. I will not give the derivation of this solution here, any book on electromagnetism will show this in detail. Basic solution:

Ex.. 2.2

Figure 2.2

The field with respect to the current-carrying conductor is cyclic and concentric.

(The direction of the magnetic lines (vectors H, B) is determined by the gimlet (corkscrew) rule. If the translational movement of the gimlet corresponds to the direction of the current in the conductor, then the rotational direction of the handle will indicate the direction of the vectors.)

Another case that has an analytical solution is the axial field of a coil with current. So far, we can get an analytical solution for the axial field, but this cannot be done for the field as a whole. To find the field at some arbitrary point, we need to solve complex integral equations, which is best done using digital methods.

3. Ampère's law

This is an alternative method for determining the magnetic field, using a group of current-carrying conductors. The law can be written as:

Ex. 3.1

where N current-carrying conductor number i and llinear vector. The integration should form a closed line around the current-carrying conductor. Considering an infinite current-carrying conductor, we can again apply Ampère's law as shown below:

Figure 3.1

We know that the field is cyclic and concentric around a current-carrying conductor, soHcan be integrated around the ring (around the conductor with current) at a distance r , which gives us:

Ex . 3.2

The integration is very simple and shows how Ampère's law can be applied to get a fast solution in some cases (configurations). Knowledge of the field structure is necessary before this law can be applied.

(Field (strength) in the center of the circular field (coil with current))

4. Solenoid field

As the charge moves in the coil, it creates a magnetic field, the direction of which can be determined using the right hand rule (take your right hand, bend your fingers in the direction of the current, bend your thumb, the direction indicated by the thumb points to the magnetic north of your coil) . The convention for magnetic flux says that magnetic flux starts at the north pole and ends at the south. ( The convention for the direction of flux has the flux emerging from a north pole and terminating on a south pole ). The field and flux lines are closed turns around the coil. Remember that these lines don't really exist, they just connect points of equal value. It's a bit like contours on a map, where the lines show points of the same height. The height of the ground changes continuously between these contours. Similarly, the field and magnetic flux are continuous (the change is not necessarily smooth - a discrete change in permeability causes a sharp change in the value of the field, a bit like rocks on a map).

Figure 4.1

If the solenoid is long and thin, then the field inside the solenoid can be considered almost uniform.

5. Ferromagnetic materials

Perhaps the most well-known ferromagnetic material is iron, but there are other elements such as cobalt and nickel, as well as numerous alloys such as silicon steel. Each material has a special property that makes it suitable for its application. So what do we mean by ferromagnetic material? It's simple, a ferromagnetic material is attracted to a magnet. While this is true, it's hardly a useful definition, and it doesn't tell us why attraction occurs. A detailed theory of the magnetism of materials is a very complex topic involving quantum mechanics, so we will stick to a simple conceptual description. As you know, the flow of charges creates a magnetic field, so when we detect the movement of a charge, we must expect an associated magnetic field. In ferromagnetic materials, the orbits of electrons are distributed in such an order that a small magnetic field is created. Then this means that the material consists of many tiny current-carrying coils, which have their own magnetic fields. Usually, coils oriented in the same direction are combined into small groups called domains. The domains point in an arbitrary direction in the material, so there is no overall magnetic field in the material (the resulting field is zero). However, if we apply an external field to the ferromagnetic material from a coil or a permanent magnet, the coils with currents turn in the direction with this field.(However if we apply an external field to the ferromagnetic material from a coil or permanent magnet, the current loops try and align with this field - the domians which are most aligned with the field "grow" at the expense of the less well aligned domains ). When this happens, the result will be magnetization and attraction between the material and the magnet/coil.

6. Magneticinductionandpermeability

Receiving a magnetic field has an associated magnetic flux density, also known as magnetic induction. InductionB connected to the field through the permeability of the medium through which the field propagates.

Ex. 6.1

where 0 is the permeability in vacuum and r relative permeability. Induction measured in teslas (T).

(The intensity of the magnetic field depends on the medium in which it occurs. Comparing the magnetic field in a wire located in a given medium and in vacuum, it was found that, depending on the properties of the medium (material), the field is stronger than in vacuum (paramagnetic materials or media ), or, conversely, weaker (diamagnetic materials and media).The magnetic properties of the medium are characterized by the absolute magnetic permeability μ a.

The absolute magnetic permeability of vacuum is called the magnetic constant μ 0 . The absolute magnetic permeability of various substances (media) is compared with the magnetic constant (magnetic permeability of a vacuum). The ratio of the absolute magnetic permeability of a substance to the magnetic constant is called magnetic permeability (or relative magnetic permeability), so that

Relative magnetic permeability is an abstract number. For diamagnetic substances μ r < 1, например для меди μ r= 0.999995. For paramagnetic substances μ r> 1, e.g. for air μ r= 1.0000031. In technical calculations, the relative magnetic permeability of diamagnetic and paramagnetic substances is assumed to be 1.

For ferromagnetic materials, which play an extremely important role in electrical engineering, the magnetic permeability has different values ​​depending on the properties of the material, the magnitude of the magnetic field, temperature and reach values tens of thousands.)

7. Magnetization

The magnetization of a material is a measure of its magnetic ‘strength’. The magnetization may be inherent in the material, such as a permanent magnet, or it may be caused by an external magnetic field source, such as a solenoid. The magnetic induction in a material can be expressed as the sum of the magnetization vectorsM and magnetic fieldH .

Ex. 7.1

(Electrons in atoms, moving along closed orbits or elementary circuits around the nucleus of an atom, form elementary currents or magnetic dipoles. The magnetic dipole can be characterized by the vector - magnetic moment dipole or elementary electric current m , the value of which is equal to the product of the elementary current i and elementary platform S , Fig.8e.0.1, limited by an elementary structure.

Rice. 8d.0.1

Vectorm directed perpendicular to the site S ; , its direction is determined by the gimlet rule. A vector quantity equal to the geometric sum of the magnetic moments of all elementary molecular currents in the body under consideration (substance volume) is magnetic moment of the body

Vector quantity determined by the ratio of the magnetic moment M ’ to volumeV , called average body magnetization or medium magnetization intensity

If the ferromagnet is not in an external magnetic field, then the magnetic moments of individual domains are directed in a very different way, so that the total magnetic moment of the body turns out to be equal to zero, i.e. ferromagnet is not magnetized. The introduction of a ferromagnet into an external magnetic field causes: 1-turn of the magnetic domains in the direction of the external field - the process of orientation; 2-an increase in the size of those domains whose moments are close to the direction of the field, and a decrease in domains with oppositely directed magnetic moments - the process of displacement of the domain boundaries. As a result, the ferromagnet is magnetized. If, with an increase in the external magnetic field, all spontaneously magnetized sections are oriented in the direction of the external field and the growth of domains stops, then the state of the limiting magnetization of the ferromagnet, called magnetic saturation.

At a field strength H, magnetic induction in a non-ferromagnetic medium (μ r= 1) would be equal to B 0 =μ 0 H. In a ferromagnetic medium, this induction is added to the induction of an additional magnetic field Bd= μ 0 M.Resulting magnetic induction in a ferromagnetic material B= B 0 + Bd=μ 0 ( H+ M).)

8. Magnetomotive force (mfs)

This is an analogue of the electromotive force (EMF) and is used in magnetic circuits to determine the magnetic flux density in different directions of the circuit. MDS measured in ampere - turns or simply in amperes . The magnetic circuit is equivalent to resistance and is called magnetic resistance, which is defined as

Ex . 8.1

where lchain path length, permeability andAcross-sectional area.

Let's take a look at a simple magnetic circuit:

Rice . 8.1

The torus has an average radius r and cross-sectional area A . MDS is generated by a coil with N coils in which current flows i . The calculation of magnetic resistance is complicated by nonlinearities in the permeability of the material.

Ex . 8.2

If the magnetic resistance is determined, then we can calculate the magnetic flux that is present in the circuit.

9. Demagnetizing fields

If a piece of ferromagnetic material, in the form of a bar, is magnetized, then poles will appear at its ends. These poles generate an internal field that tries to demagnetize the material - it acts in the opposite direction of the field that creates the magnetization. As a result, the internal field will be much smaller than the external one. The shape of the material matters a lot to the demagnetizing field, a long thin rod (large length/diameter ratio) has a small demagnetizing field compared to, say, a wide shape like a sphere. In the development perspective coilgun this means that a projectile with a small length/diameter ratio requires a stronger external field to achieve a certain state of magnetization. Take a look on the chart below. It shows the resulting internal field along the axis of two projectiles - one 20 mm long and 10 mm in diameter and the other 10 mm long and 20 mm in diameter. For the same external field, we see a big difference in internal fields, the shorter projectile has a peak of about 40% of the long projectile's peak. This is a very successful result, showing the difference between different forms of projectiles.

Rice . 9.1

It should be noted that poles are formed only where there is a continuous permeability of the material. On a closed magnetic path, like a torus, poles do not arise, and there is no demagnetizing field.

10. Force acting on a charged particle

So, how do we calculate the force acting on a conductor with current? Let's start by looking at the force acting on a charge moving in a magnetic field. ( I "ll adopt the general approach in 3 dimensions ).

Ex . 10.1

This force is determined by the intersection of the velocity vectorsvand magnetic inductionB, and it is proportional to the magnitude of the charge. Consider the charge q = -1.6x 10 -19 K, moving at a speed of 500m/s in a magnetic field with an induction of 0.1 T l as shown below.

Rice . 10.1. The effect of a force on a moving charge

The force experienced by the charge can be simply calculated as shown below:

Speed ​​vector 500i m/s and induction 0.1 k T , so we have:

Obviously, if nothing resists this force, the particle willdeviate (it will have to describe a circle in the plane x-y for the case above). There are many interesting special cases that can be obtained with free charges and magnetic fields - you have only read about one of them.

11. Force acting on a conductor with current

Now let's refer what we learned to the force acting on a conductor with current. There is two different ways to get the ratio .

We can describe the conditional current as a measure of charge change

Ex . 11.1

Now we can differentiate the force equation given above to get

Ex. 11.2

We combine these equations, we get

Ex. 11.3

d l is a vector showing the direction of the conditional current. The expression can be used to analyze a physical organization such as a DC motor. If a the conductor is straight, then this can be simplified to

Ex. 11.4

The direction of the force always creates a right angle to the magnetic flux and the direction of the current. When is the simplified form used?, the direction of the force is determined by the right hand rule.

12. Induced voltage, Faraday's law, Lenz's law

The last thing we need to consider is the induced voltage. This is simply an extended analysis of the effect of a force on a charged particle. If we take a conductor (something with a mobile charge) and give it some speed V , relative to the magnetic field, a force will act on the free charges, which pushes them to one of the ends of the conductor. In a metal bar there will be a charge separation where the electrons will be collected at one of the ends of the bar. Picture below shows the general idea.

Rice. 12.1 Induced voltage during transverse movement of the conductive bar

Any relative motion between the conductor and the induction of the magnetic field will result in an induced voltage generated by the movement of the charges. However, if the conductor moves parallel to the magnetic flux (along the axis Z in the figure above), then no voltage will be induced.

We can consider another situation where an open planar surface is pierced by a magnetic current. If we place a closed loop there C , then any change in magnetic flux associated with C will generate tension around C.

Rice . 12.2 Magnetic flux associated with the circuit

Now if we imagine the conductor as a closed loop in place C , then a change in the magnetic flux will induce a voltage in this conductor, which will move the current in a circle in this coil. The direction of the current can be determined by applying Lenz's law, which, in simple terms, shows that the result of the action is directed opposite to the action itself. In this case, the induced voltage will drive a current which will prevent the magnetic flux from changing - if the magnetic flux decreases then the current will try to keep the magnetic flux unchanged (counterclockwise), if the magnetic flux increases then the current will prevent this increase (clockwise ) (the direction is determined by the gimlet rule) . Faraday's law establishes the relationship between induced voltage, change in magnetic flux, and time:

Eqn 12.1

Minus takes into account Lenz's law.

13. Inductance

Inductance can be described as the ratio of the associated magnetic flux to the current that this magnetic flux creates. For example, consider a coil of wire with a cross-sectional area A in which flows I.

Rice. 13.1

The inductance itself can be defined as

Eqn 13.1

If there is more than one turn then the expression becomes

Eqn 13.2

where N- number of turns.

It is important to understand that inductance is only a constant if the coil is surrounded by air. When a ferromagnetic material appears as part of a magnetic circuit, then there is a non-linear behavior of the system, which gives a variable inductance.

14. transformationelectromechanical energy

The principles of electromechanical energy conversion apply to all electrical machines and coilgun not an exception. Before consideration coilgun let's imagine a simple linear electrical 'motor' consisting of a stator field and an armature placed in this field. This is shown in fig. 14.1. Note that in this simplified analysis, the voltage source and armature current do not have an inductance associated with them. This means that only the induced voltage in the system is a consequence of the movement of the armature with respect to the magnetic induction.

Rice. 14.1. Primitive linear motor

When voltage is applied to the ends of an armature, the current will be determined according to its resistance. This current will experience a force ( I x B ), causing the anchor to accelerate. Now, using the previously discussed section ( 12 Induced voltage, Faraday's law, Lenz's law ), we have shown the fact that a voltage is induced in a conductor moving in a magnetic field. This induced voltage acts opposite to the applied voltage (according to Lenz's law). Rice. 14.2 shows an equivalent circuit in which electrical energy is converted into thermal energy P T , and mechanical energy P M .

Rice . 14.2. Motor equivalent circuit

Now we need to consider how the mechanical energy of the armature relates to the electrical energy transmitted to it. Since the armature is located at right angles to the field of magnetic induction, the force is determined by the simplified expression 1 1.4

Ex . 14.1

since instantaneous mechanical energy is a product of force and speed, we have

Ex . 14.2

where v- anchor speed. If we apply Kirchhoff's law to a closed circuit, we get the following expressions for the current I.

Ex . 14.3

Now, the induced voltage can be expressed as a function of the armature speed

Ex . 14.4

Substituting vyp . 14.4 in 1 4.3 we get

Ex . 14.5

and substituting vyp.14.5 in 14.2 we get

Ex . 14.6

Now let's consider the thermal energy released in the anchor. It is determined by vyp. 14.7

Ex . 14.7

And finally, we can express the energy supplied to the anchor as

Ex . 14.8

Note also that mechanical energy (vyp.14.2) is the equivalent of current I multiplied by the induced voltage (vyr.14.4).

We can plot these curves to see how the energy delivered to the anchor is combined with the speed range.(We can plot these curves to show how the power supplied to the armature is distributed over a range of speeds).For this analysis to be relevant to coilgun , we will give our variables values ​​that match the accelerator coilgun . Let's start with the current density in the wire, from which we will determine the values ​​of the remaining parameters. The maximum current density during testing was 90 A /mm 2 , so if we choose the length and diameter of the wire as

l = 10 m

D = 1.5x10 -3 m

then the resistance of the wire and the current will be

R = 0.1

I = 160A

Now we have values ​​for resistance and current, we can determine the voltage

V=16V

All these parameters are necessary to build the static characteristics of the motor.

Rice. 14.3 Performance curves for frictionless motor model

We can make this model a little more realistic by adding a friction force of, say, 2N, so that the reduction in mechanical energy is proportional to the armature speed. The value of this friction is deliberately taken more in order to make the effect of this more obvious. The new set of curves is shown in Figure 14.4.

Rice . 14.4. Performance curves with constant friction

The presence of friction slightly changes the energy curves so that the maximum armature speed is slightly less than in the case of zero friction. The most noticeable difference is the change in the efficiency curve, which now peaks and then drops off sharply when the anchor reaches " no-load speed. This shape of the efficiency curve is typical for a permanent magnet DC motor.

Also noteworthy is how force, and hence acceleration, depends on speed. If we substitute ex.14.5 into ex.14.1 we get an expression for F in terms of speed v.

Ex . 14.9

Having built this dependence, we will get the following graph

Rice. 14.5. The dependence of the force acting on the anchor on the speed

It is clear that the armature starts with the maximum accelerating force, which begins to decrease as soon as the armature begins to move. Although these characteristics give instantaneous values ​​of the actual parameters for a certain speed, they should be useful in order to see how the motor behaves over time, i.e. dynamically.

The dynamic response of a motor can be determined by solving a differential equation that describes its behavior. Rice. 14.6 shows a diagram of the effect of forces on the anchor, from which you can determine the resulting force described by the differential equation.

Rice. 14.6 Diagram of the effect of forces on the anchor

F m and F d are the magnetic and opposing forces, respectively. Since stress is a constant value, we can use Eq. 14.1 and the resulting force Fa , acting on the anchor, will be

. 14.11

If we write acceleration and speed as derivatives of displacement x with respect to time and rearrange the expression , we get differential equation for motion anchors

vyr. 14.12

This is a non-homogeneous second-order differential equation with constant coefficients and can be solved by defining an additional function and a partial integral. The method of solving a straight line (all programs of mathematical universities consider differential equations), so I will simply give the result. One note - this particular solution uses initial conditions:

vyr. 14.14

We need to assign a value to the friction force, magnetic induction and armature mass. Let's choose friction. I will use the 2H value to illustrate how it changes the dynamic performance of the motor. Determining the value of induction that will produce the same accelerating force in the model as it does in the test coil for a given current density requires that we consider the radial component of the magnetic flux density distribution generated by the magnetized projectile.coilgun(this radial component creates an axial force). To do this, it is necessary to integrate the expression obtained by multiplying the current densityDetermining the volume integral of the radial magnetic flux density usingFEMM

The projectile becomes magnetized when we define for itB- Hcurve andhcvalues ​​inFEMMmaterial properties dialog. Valueswerechosenforstrictcompliancewithmagnetizediron. FEMMgives a value of 6.74x10 -7 Tm 3 for the volume integral of the magnetic flux densityB coil, so usingF= /4 we getB model = 3.0 x10 -2 Tl. This value of magnetic flux density may seem very small, considering the magnetic flux density inside the projectile, which is somewhere around 1.2Tl, however, we must understand that the magnetic flux unfolds in a much larger volume around the projectile with only a portion of the magnetic flux shown in the radial component. Now you understand that, according to our model,coilgun- This "insideout"(turned inside out) and "backtofront", in other words,coilgunthe immovable copper surrounds the magnetized part, which is moving. This does not create any problems. So the essence of the system is the connected linear force acting on the stator and the armature, so we can fix the copper part and allow the stator field to create movement. The stator field generator is our shell, let's assign a mass of 12g to it.

We can now plot displacement and velocity as a function of time, as shown in Fig. 14.8

Rice. 14.8. Dynamic behavior of a linear motor

We can also combine the expressions for velocity and displacement to obtain a function of velocity from displacement, as shown in Fig. 14.9.

Rice. 14.9. Characteristic of the dependence of speed on displacement

It is important to note here that a relatively long accelerator is needed for the anchor to begin to reach its maximum speed. This isIt hasmeaningforbuildingmaximum efficientpracticalaccelerator.

If we enlarge the curves, we can determine what speed will be reached at a distance equal to the length of the active material in the spool of the accelerator gun (78 mm).

Rice. 14.10. Increased speed versus displacement curve

These are remarkably close to those of the actual 3-stage accelerator, however this is just a coincidence as there are several significant differences between this model and the actualcoilgun. For example, incoilgunforce is a function of speed and displacement coordinates, and in the presented model, force is only a function of speed.

Rice. 14.11 - dependence of the total efficiency of the motor as a projectile accelerator.

Rice. 14.11. Cumulative Efficiency as a Function of Displacement Without Friction Loss

Rice. 14.11. Cumulative Efficiency as a Function of Displacement Considering Constant Friction Losses

The cumulative efficiency shows a fundamental feature of this type of electrical machine - energy is acquired by the armature when it accelerates first and up to ‘no- load’ speed is exactly half of the total energy delivered to the car. In other words, the maximum possible efficiency of an ideal (frictionless) accelerator would be 50%. If there is friction, then the cumulative efficiency will show the maximum efficient point that occurs due to the operation of the machine against friction.

Finally, let's look at the impactBon the dynamic characteristics of speed-displacement, as shown in Figures 14.10 and 14.11.

Rice. 14.11. InfluenceBon gradient speed-displacement

Rice. 14.12. Area of ​​small displacement where increasing induction gives more speed

This set of curves shows an interesting feature of this model, in which a large inductance of the field in the initial stage gives a greater speed at a particular point, but as the speed increases, the curves corresponding to the lower inductance catch up with this curve. This explains the following: You decide that a higher induction will give a greater initial acceleration, however, in accordance with the fact that a larger induced voltage will be induced, the acceleration will decrease more sharply, allowing the curve for the lower induction to catch up with this curve.

So what have we learned from this model? I think the important thing to understand is that starting from a standstill, the efficiency of such a motor is very low, especially if the motor is short. The instantaneous efficiency increases once the projectile picks up speed due to the induced voltage reducing the current. This increases the efficiency because the energy loss in the resistance (obviously heat loss) decreases and the mechanical energy increases (see figs 14.3, 14.4), however, since the acceleration also falls, a progressively larger displacement is obtained, so the best efficiency curve will be used.(In short, a linear motor subjected to a step voltage "forcing function" is going to be quite an inefficient machine unless it is very long.)

This primitive motor model is useful in that it shows a case of typical weak efficiencycoilgun, namely the low level driving induced voltage. The model is simplified and does not take into account the non-linear and inductive elements of the practical circuit, therefore, in order to enrich the model, we need to include these elements in our electrical model circuit. In the next section, you will learn the basic differential equations for a single-stagecoilgun. In the analysis, we will try to obtain an equation that could be solved analytically (with the help of several simplifications). If this fails, I will use Runge Kutta's numerical integration algorithm.

There are four fundamental forces of physics, and one of them is called electromagnetism. Ordinary magnets are of limited use. An electromagnet is a device that creates during the passage of an electric current. Since electricity can be turned on and off, the same goes for an electromagnet. It can even be weakened or strengthened by reducing or increasing the current. Electromagnets find their application in a variety of everyday electrical appliances, in various industries, from conventional switches to spacecraft propulsion systems.

What is an electromagnet?

An electromagnet can be thought of as a temporary magnet that functions with the flow of electricity and its polarity can be easily changed by changing Also the strength of an electromagnet can be changed by changing the amount of current flowing through it.

The scope of electromagnetism is unusually wide. For example, magnetic switches are preferred because they are less susceptible to temperature changes and are able to maintain rated current without nuisance tripping.

Electromagnets and their applications

Here are some of the examples where they are used:

• Motors and generators. Thanks to electromagnets, it became possible to manufacture electric motors and generators that operate on the principle of electromagnetic induction. This phenomenon was discovered by the scientist Michael Faraday. He proved that electric current creates a magnetic field. The generator uses the external force of the wind, moving water or steam to rotate a shaft that causes a set of magnets to move around a coiled wire to create an electrical current. Thus, electromagnets convert other types of energy into electrical energy.
• The practice of industrial use. Only materials made from iron, nickel, cobalt or their alloys, as well as some natural minerals, react to a magnetic field. Where are electromagnets used? One area of ​​practical application is the sorting of metals. Since these elements are used in production, iron-containing alloys are effectively sorted using an electromagnet.
• Where are electromagnets used? They can also be used to lift and move massive objects such as cars before scrapping. They are also used in transportation. Trains in Asia and Europe use electromagnets to carry cars. This helps them move at phenomenal speeds.

Electromagnets in everyday life

Electromagnets are often used to store information, as many materials are capable of absorbing a magnetic field that can later be read to extract information. They find application in almost any modern device.

Where are electromagnets used? In everyday life, they are used in a number of household appliances. One of the useful characteristics of an electromagnet is the ability to change when changing the strength and direction of the current flowing through the coils or windings around it. Speakers, loudspeakers, and tape recorders are devices in which this effect is realized. Some electromagnets can be very strong, and their strength can be regulated.

Where are electromagnets used in life? The simplest examples are electromagnetic locks. An electromagnetic interlock is used for the door, creating a strong field. As long as current flows through the electromagnet, the door remains closed. Televisions, computers, cars, elevators and copiers are where electromagnets are used, and this is by no means a complete list.

Electromagnetic forces

The strength of the electromagnetic field can be controlled by varying the electric current passing through the wires wrapped around the magnet. If you change the direction of the electric current, the polarity of the magnetic field is also reversed. This effect is used to create fields in magnetic tape or a computer hard drive for storing information, as well as in the loudspeakers of speakers in radio, television and stereo systems.

Magnetism and electricity

Dictionary definitions of electricity and magnetism differ, although they are manifestations of the same force. When electric charges move, they create a magnetic field. Its change, in turn, leads to the appearance of an electric current.

Inventors use electromagnetic forces to create electric motors, generators, toy machines, consumer electronics and many other invaluable devices, without which it is impossible to imagine the daily life of a modern person. Electromagnets are inextricably linked with electricity, they simply cannot work without an external power source.

Application of lifting and large-scale electromagnets

Electric motors and generators are vital in today's world. The motor takes in electrical energy and uses a magnet to turn the electrical energy into kinetic energy. A generator, on the other hand, converts motion using magnets to generate electricity. When moving large metal objects, lifting electromagnets are used. They are also necessary when sorting scrap metal, for separating cast iron and other ferrous metals from non-ferrous ones.

A real miracle of technology is a Japanese levitating train capable of reaching speeds of up to 320 kilometers per hour. It uses electromagnets to help it float in the air and move incredibly fast. The US Navy is conducting high-tech experiments with a futuristic electromagnetic railgun. She can direct her projectiles over considerable distances at great speed. The projectiles have enormous kinetic energy, so they can hit targets without the use of explosives.

The concept of electromagnetic induction

In the study of electricity and magnetism, the concept is important when a flow of electricity occurs in a conductor in the presence of a changing magnetic field. The use of electromagnets with their inductive principles is actively used in electric motors, generators and transformers.

Where can electromagnets be used in medicine?

Magnetic resonance imaging (MRI) scanners also work with electromagnets. This is a specialized medical method for examining human internal organs that are not available for direct examination. Along with the main one, additional gradient magnets are used.

Where are electromagnets used? They are present in all kinds of electrical devices, including hard drives, speakers, motors, generators. Electromagnets are used everywhere and, despite their invisibility, occupy an important place in the life of modern man.

Lecture plan

1. Electrostatics. Short review.

2. Magnetic interaction of electric currents.

3. Magnetic field. Ampere's law. Magnetic field induction.

4. Biot-Savart-Laplace law. The principle of superposition of magnetic fields.

4.1. Magnetic field of rectilinear current.

4.2. Magnetic field on the axis of circular current.

4.3. The magnetic field of a moving charge.

1. Electrostatics. Short review.

Let us preface the study of magnetostatics with a brief review of the main provisions of electrostatics. Such an introduction seems appropriate, because when creating the theory of electromagnetism, methodological techniques were used that we have already met in electrostatics. That is why it is not superfluous to remember them.

1) The main experimental law of electrostatics - the law of interaction of point charges - Coulomb's law:

Immediately after its discovery, the question arose: how do point charges interact at a distance?

Coulomb himself adhered to the concept of long-range action. However, Maxwell's theory and subsequent experimental studies of electromagnetic waves showed that the interaction of charges occurs with the participation of electric fields created by charges in the surrounding space. Electric fields are not an ingenious invention of physicists, but an objective reality of nature.

2) The only manifestation of an electrostatic field is the force acting on a charge placed in this field. Therefore, there is nothing unexpected in the fact that the main characteristic of the field is the intensity vector associated with this particular force:

,. (E2)

3) Combining the definition of tension (E2) and Coulomb's law (E1), we find the field strength created by one point charge:

. (E3)

4) Now - very important experienced result: principle of superposition of electrostatic fields:

. (E4)

This "principle" made it possible to calculate the electric fields created by charges of a wide variety of configurations.

With this, perhaps, we can limit our brief review of electrostatics and move on to electromagnetism.

1. Magnetic interaction of electric currents

The interaction of currents was discovered and studied in detail by Ampère in 1820.

On fig. 8.1. a diagram of one of his experimental setups is given. Here, the rectangular frame 1 has the ability to easily rotate around a vertical axis. Reliable electrical contact when turning the frame was provided by mercury poured into the support cups. If another frame with current (2) is brought to such a frame, then an interaction force arises between the near sides of the frames. It was this force that Ampère measured and analyzed, considering that the interaction forces of the distant edges of the frames can be neglected.

Rice. 8.1.

Experimentally, Ampere established that parallel currents of the same direction (Fig. 8.2., a), interacting, attract, and oppositely directed currents repel (Fig. 8.2., b). When parallel currents interact, a force acts per unit length of the conductor, which is proportional to the product of the currents and inversely proportional to the distance between them ( r):

. (8.1)

Rice. 8.2.

This experimental law of the interaction of two parallel currents is used in the SI system to determine the basic electrical unit - the unit of current strength 1 ampere.

1 ampere is the strength of such a direct current, the flow of which along two straight conductors of infinite length and small cross section, located at a distance of 1 m from each other in a vacuum, is accompanied by the appearance between the conductors of a force equal to 2 10 –7 H for each meter of their length.

Having thus determined the unit of current strength, we find the value of the proportionality coefficient  in expression (8.1):

.

At I 1 =I 2 = 1A and r = 1 m force acting on each meter of conductor length
= 210 –7 N/m. Hence:

.

In rationalized SI = , where  0 - magnetic constant:

 0 = 4= 410 –7
.

For a very short time, the nature of the force interaction of electric currents remained unclear. In the same 1820, the Danish physicist Oersted discovered the effect of electric current on a magnetic needle (Fig. 8.3.). In Oersted's experiment, a straight conductor was stretched over a magnetic needle oriented along the Earth's magnetic meridian. When the current is turned on in the conductor, the arrow rotates, setting itself perpendicular to the current-carrying conductor.

Rice. 8.3.

This experiment directly indicates that the electric current creates a magnetic field in the surrounding space. Now we can assume that the ampere force of the interaction of currents has an electromagnetic nature. It arises as a result of the action on the electric current of the magnetic field created by the second current.

In magnetostatics, as well as in electrostatics, we have come to the field theory of the interaction of currents, to the concept of short-range action.

The first law of electromagnetism describes the flow of an electric field:

where e 0 is some constant (read epsilon zero). If there are no charges inside the surface, but there are charges outside it (even very close to it), then all the same average the normal component of E is zero, so there is no flow through the surface. To show the usefulness of this type of statement, we will prove that equation (1.6) coincides with Coulomb's law, if only we take into account that the field of an individual charge must be spherically symmetric. Draw a sphere around a point charge. Then the average normal component is exactly equal to the value of E at any point, because the field must be directed along the radius and have the same magnitude at all points on the sphere. Our rule then states that the field on the surface of the sphere times the area of ​​the sphere (i.e., the flux flowing out of the sphere) is proportional to the charge inside it. If you increase the radius of a sphere, then its area increases as the square of the radius. The product of the average normal component of the electric field and this area must still be equal to the internal charge, so the field must decrease as the square of the distance; thus the field of "inverse squares" is obtained.

If we take an arbitrary curve in space and measure the circulation of the electric field along this curve, then it turns out that in the general case it is not equal to zero (although this is the case in the Coulomb field). Instead, the second law holds for electricity, stating that

And, finally, the formulation of the laws of the electromagnetic field will be completed if we write two corresponding equations for the magnetic field B:

And for the surface S, bounded curve WITH:

The constant c 2 that appeared in equation (1.9) is the square of the speed of light. Its appearance is justified by the fact that magnetism is essentially a relativistic manifestation of electricity. And the constant e o was set in order for the usual units of electric current strength to arise.

Equations (1.6) - (1.9), as well as equation (1.1) - these are all the laws of electrodynamics.

As you remember, Newton's laws were very easy to write, but many complex consequences followed from them, so it took a long time to study them all. The laws of electromagnetism are incomparably more difficult to write, and we must expect the consequences of them to be much more complicated, and now we will have to understand them for a very long time.

We can illustrate some of the laws of electrodynamics with a series of simple experiments that can show us at least qualitatively the relationship between electric and magnetic fields. You get to know the first term in equation (1.1) by combing your hair, so we won't talk about it. The second term in equation (1.1) can be demonstrated by passing a current through a wire suspended over a magnetic bar, as shown in Fig. 1.6. When the current is turned on, the wire moves due to the fact that a force F = qvXB acts on it. When a current flows through the wire, the charges inside it move, that is, they have a speed v, and the magnetic field of the magnet acts on them, as a result of which the wire moves away.

When the wire is pushed to the left, the magnet itself can be expected to experience a push to the right. (Otherwise, this whole device could be mounted on a platform and get a reactive system in which momentum would not be conserved!) Although the force is too small to notice the movement of a magnetic wand, the movement of a more sensitive device, say a compass needle, is quite noticeable.

How does the current in the wire push the magnet? The current flowing through the wire creates its own magnetic field around it, which acts on the magnet. In accordance with the last term in equation (1.9), the current should lead to circulation vector B; in our case, field lines B are closed around the wire, as shown in fig. 1.7. It is this field B that is responsible for the force acting on the magnet.

Fig.1.6. Magnetic stick that creates a field near the wire AT.

When current flows through the wire, the wire is displaced due to the force F = q vxb.

Equation (1.9) tells us that for a given amount of current flowing through the wire, the circulation of the field B is the same for any curve surrounding the wire. Those curves (circles, for example) that lie far from the wire have a longer length, so the tangent component B must decrease. You can see that B should be expected to decrease linearly with distance from a long straight wire.

We said that the current flowing through the wire forms a magnetic field around it, and that if there is a magnetic field, then it acts with some force on the wire through which the current flows.

Fig.1.7. The magnetic field of the current flowing through the wire acts on the magnet with some force.

Fig. 1.8. Two wires carrying current

also act on each other with a certain force.

So, one should think that if a magnetic field is created by a current flowing in one wire, then it will act with some force on the other wire, through which the current also flows. This can be shown by using two freely suspended wires (Fig. 1.8). When the direction of the currents is the same, the wires attract, and when the directions are opposite, they repel.

In short, electric currents, like magnets, create magnetic fields. But then what is a magnet? Since magnetic fields are created by moving charges, can it not turn out that the magnetic field created by a piece of iron is actually the result of the action of currents? Apparently, that's the way it is. In our experiments it is possible to replace the magnetic stick with a coil of wound wire, as shown in Fig. 1.9. When the current passes through the coil (as well as through a straight wire above it), exactly the same movement of the conductor is observed as before, when a magnet was in place of the coil. Everything looks as if a current circulates continuously inside a piece of iron. Indeed, the properties of magnets can be understood as a continuous current within the iron atoms. The force acting on the magnet in Fig. 1.7 is explained by the second term in equation (1.1).

Where do these currents come from? One source is the movement of electrons in atomic orbits. In iron this is not the case, but in some materials the origin of magnetism is precisely this. In addition to rotating around the nucleus of an atom, the electron also rotates around its own axis (something similar to the rotation of the Earth); it is from this rotation that a current arises, which creates a magnetic field of iron. (We said "something like the rotation of the Earth" because, in fact, the matter in quantum mechanics is so deep that it does not fit well into classical concepts.) In most substances, some electrons spin in one direction, some in the other, so that magnetism disappears, and in iron (for a mysterious reason, which we will discuss later) many electrons rotate so that their axes point in the same direction and this is the source of magnetism.

Since the fields of magnets are generated by currents, there is no need to insert additional terms into equations (1.8) and (1.9) that take into account the existence of magnets. These equations are about all currents, including circular currents from rotating electrons, and the law turns out to be correct. It should also be noted that, according to equation (1.8), there are no magnetic charges similar to electric charges on the right side of equation (1.6). They have never been discovered.

The first term on the right side of equation (1.9) was discovered theoretically by Maxwell; he is very important. He says change electrical fields causes magnetic phenomena. In fact, without this term, the equation would lose its meaning, because without it the currents in open circuits would disappear. But in fact, such currents exist; the following example speaks of this. Imagine a capacitor made up of two flat plates.

Fig. 1.9. The magnetic stick shown in Fig. 1.6

can be replaced by a coil that flows

The force will still be acting on the wire.

Fig. 1.10. The circulation of the field B along the curve C is determined either by the current flowing through the surface S 1 or by the rate of change of the flow, the field E through the surface S 2 .

It is charged by current flowing into one of the plates and outflowing from the other, as shown in Fig. 1.10. Draw a curve around one of the wires With and stretch a surface over it (surface S 1 , that crosses the wire. In accordance with equation (1.9), the circulation of the field B along the curve With is given by the amount of current in the wire (multiplied by from 2). But what happens if we pull on the curve another surface S 2 in the form of a cup, the bottom of which is located between the plates of the capacitor and does not touch the wire? No current passes through such a surface, of course. But a simple change in the position and shape of an imaginary surface should not change the real magnetic field! The circulation of field B must remain the same. Indeed, the first term on the right-hand side of equation (1.9) is combined with the second term in such a way that for both surfaces S 1 and S 2 the same effect occurs. For S2 the circulation of the vector B is expressed in terms of the degree of change in the flow of the vector E from one plate to another. And it turns out that the change in E is connected with the current just so that equation (1.9) is satisfied. Maxwell saw the need for this and was the first to write the complete equation.

With the device shown in Fig. 1.6, another law of electromagnetism can be demonstrated. Disconnect the ends of the hanging wire from the battery and attach them to a galvanometer - a device that records the passage of current through the wire. Stands only in the field of a magnet swing wire, as the current will immediately flow through it. This is a new consequence of equation (1.1): the electrons in the wire will feel the action of the force F=qvXB. Their speed is now directed to the side, because they deviate along with the wire. This v, together with the vertically directed field B of the magnet, results in a force acting on the electrons along wires, and the electrons are sent to the galvanometer.

Let us suppose, however, that we leave the wire alone and begin to move the magnet. We feel that there should be no difference, because the relative motion is the same, and indeed the current flows through the galvanometer. But how does a magnetic field act on charges at rest? In accordance with equation (1.1), an electric field should arise. A moving magnet must create an electric field. The question of how this happens is answered quantitatively by equation (1.7). This equation describes many practically very important phenomena occurring in electrical generators and transformers.

The most remarkable consequence of our equations is that, by combining equations (1.7) and (1.9), one can understand why electromagnetic phenomena propagate over long distances. The reason for this, roughly speaking, is something like this: suppose that somewhere there is a magnetic field that increases in magnitude, say, because a current is suddenly passed through the wire. Then it follows from equation (1.7) that the circulation of the electric field should occur. When the electric field begins to gradually increase for circulation to occur, then, according to equation (1.9), magnetic circulation must also occur. But the rise this the magnetic field will create a new circulation of the electric field, etc. In this way, the fields propagate through space, requiring neither charges nor currents anywhere but the source of the fields. It is in this way that we see each other! All this is hidden in the equations of the electromagnetic field.

End of work -

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straight wire
As a first example, let us again calculate the field of a straight wire, which we found in the previous paragraph, using equation (14.2) and symmetry considerations. Take a long straight wire

Long solenoid
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Small loop field; magnetic dipole
Let's use the vector potential method to find the magnetic field of a small loop with current. As usual, by the word "small" we simply mean that we are only interested in fields on a large scale.

Vector circuit potential
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Broadcasting

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Magnetotherapy

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Synchrophasotrons

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In the generator mode, the armature of the machine rotates under the influence of an external moment. Between the poles of the stator there is a constant magnetic flux penetrating the armature. The armature winding conductors move in a magnetic field and, therefore, an EMF is induced in them, the direction of which can be determined by the "right hand" rule. In this case, a positive potential arises on one brush relative to the second. If a load is connected to the generator terminals, then current will flow in it.

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Transformers are widely used in the transmission of electrical energy over long distances, its distribution between receivers, as well as in various rectifying, amplifying, signaling and other devices.

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If three times more turns are wound in the secondary winding of the transformer than in the primary, then the magnetic field created in the core by the primary winding, crossing the turns of the secondary winding, will create three times more voltage in it.

Using a transformer with a reverse turns ratio, you can just as easily and simply get a reduced voltage.