Laws of electromagnetism in technology. Laws of Electromagnetism - Miracles of Ordinary Things

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

Basics of Electromagnetism

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

Electromagnetic fieldsAndstrength

Before we consider a special case - coilgun -ah, 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 current in a conductor, rotation of an electron in its orbit, plasma flow, etc. To make electromagnetism easier to understand, we use the concept electromagnetic field and magnetic poles. Differential vector equations, which 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 are called Sommerfield, Kennely and Gaussian . Since each system has different elements (names) for many of the same things, it can get confusing. I will use Sommerfield The system shown below:

Table 1. Sommerfield Measuring system

2. LawBio- Savara

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

Fig 2.1

Where H field component at a distance r , created by current i , flowing in an elementary section of a 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 through which current flows i . We can use Biot-Savart's law to obtain the basic solution for the field at any distance from the conductor. I will not give this solution here; any book on electromagnetism will show this in detail. Basic solution:

Figure 2.2

The field in relation to the current-carrying conductor is cyclic and concentric.

(Direction magnetic lines(vectors H, B) is determined by the gimlet (corkscrew) rule. If forward movement 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. While we can obtain an analytical solution for the axial field, this cannot be done for the field as a whole. To find a field in some arbitrary point we need to solve complex integral equations, which is best done using digital methods.

3. Ampere's law

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

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

Fig 3.1

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

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

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

4. Solenoid field

When a charge moves in a 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 thumb, the direction pointed by your thumb points to the magnetic north of your coil). Agreement for magnetic flux says that magnetic flux starts from north pole and ends in 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 magnetic flux lines are closed turns around the coil. Remember that these lines don't actually exist, they just connect dots. equal value. This is slightly reminiscent of contours on a map, where the lines show points of equal height. The height of the ground changes continuously between these contours. Also the field and magnetic flux are continuous (the change is not necessarily smooth - a discrete change in permeability causes an abrupt change in the value of the field, a bit like rocks on a map).

Fig 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, ferromagnetic material is attracted to a magnet. Even though this is true, it is hardly a useful definition, and it does not tell us why attraction occurs. The detailed theory of magnetism of materials is very complex topic, which includes 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 should expect an associated magnetic field. In ferromagnetic materials, the electron orbits are distributed in such an order that a small magnetic field is created. This then means that the material consists of many tiny current-carrying coils that have their own magnetic fields. Typically, turns oriented in one direction are grouped into small groups called domains. The domains are directed in any direction in the material, so there is no net magnetic field in the material (the resulting field is zero). However, if we apply an external field to a ferromagnetic material from a coil or permanent magnet, the coils with currents turn in the direction of 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

The production of 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.

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

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

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

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, for example for air μ r= 1.0000031. For technical calculations, the relative magnetic permeability of diamagnetic and paramagnetic substances is assumed to be equal to 1.

In ferromagnetic materials that play exclusively important role in electrical engineering, magnetic permeability has different values ​​depending on the properties of the material, the magnitude of the magnetic field, temperature and reaches values tens of thousands.)

7. Magnetization

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

(Electrons in atoms, moving in closed orbits or elementary contours around the nucleus of an atom, form elementary currents or magnetic dipoles. A magnetic dipole can be characterized by a vector – magnetic moment dipole or elementary electric current m , whose value is equal to the product of the elementary current i and elementary site S , Fig. 8d.0.1, limited by an elementary cluster.

Rice. 8d.0.1

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

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

If the ferromagnet is not in an external magnetic field, then the magnetic moments of the individual domains are directed in very different ways, so that the total magnetic moment of the body turns out to be zero, i.e. a ferromagnet is not magnetized. The introduction of a ferromagnet into an external magnetic field causes: 1-rotation of magnetic domains in the direction of the external field - the orientation process; 2-increase in the size of those domains whose moment directions are close to the direction of the field, and a decrease in domains with opposite directions magnetic moments– the process of shifting domain boundaries. As a result, the ferromagnet becomes magnetized. If, with an increase in the external magnetic field, all spontaneously magnetized areas are oriented in the direction external field and the growth of domains stops, then a state of extreme magnetization of the ferromagnet will occur, called magnetic saturation.

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

8. Magnetomotive force (MF)

It is analogous to 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 amperes. A magnetic circuit is equivalent to resistance and is called magnetic reluctance, which is defined as

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 . MMF is generated by a coil with N turns in which current flows i . The calculation of magnetic resistance is complicated by nonlinearities in the permeability of the material.

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

9. Demagnetizing fields

If a piece of ferromagnetic material, shaped like a bar, is magnetized, then poles will appear at its ends. These poles generate an internal field that attempts to demagnetize the material - it acts in the opposite direction to 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 has great value to a 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 future development coilgun this means that a projectile with a small length/diameter ratio requires a stronger external field to achieve a certain magnetization state. 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 about 40% of the long projectile's peak. This is a very good result, showing the difference between different projectile shapes.

Rice . 9.1

It should be noted that poles only form where there is continuous permeability of the material. On a closed magnetic path, like a torus, no poles arise, and there is no demagnetizing field.

10. Force acting on a charged particle

So, how do we calculate the force acting on a current-carrying conductor? 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).

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

Rice . 10.1. Effect of 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 from free charges and magnetic fields - you only read about one of them.

11. Force acting on a current-carrying conductor

Now let's relate what we learned to the force acting on a current-carrying conductor. Eat two different ways to obtain the ratio.

We can describe the conventional current as an indicator of the change in charge

We can now differentiate the force equation given above to get

Let's combine these equations, we get

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

The direction of the force always creates a right angle to the magnetic flux and the direction of the current. When to use the simplified form, 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 induced voltage. This simply an extended analysis of the effect of 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 into one of the ends of the conductor. In a metal bar there will be a separation of charges where the electrons will be collected at one end of the bar. Drawing below shows the general idea.

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

The result of any relative motion between the conductor and the induction of the magnetic field will be 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 threaded magnetic current. If we put a closed loop there C , then any change in the magnetic flux associated with C will create tension around C.

Rice . 12.2 Magnetic flux coupled by a loop

Now, if we imagine the conductor as a closed coil in place C , then the change in magnetic flux will induce a voltage in this conductor, which will move the current in a circle in this turn. The direction of the current can be determined by applying Lenz's law, which, simply put, shows that the result of an effect is in the opposite direction of the effect 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 constant (counterclockwise), if the magnetic flux increases, then the current will prevent this increase (clockwise) ) (direction determined by gimlet rule) . Faraday's law establishes the relationship between induced voltage, change in magnetic flux, and time:

The 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 turn of wire with a cross-sectional area A , in which it flows I.

Rice. 13.1

The inductance itself can be defined as

If there is more than one turn, then the expression becomes

Where N – number of turns.

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

14. Conversionelectromechanical energy

The principles of electromechanical energy conversion apply to all electrical machines and coilgun no exception. Before consideration coilgun Let's imagine a simple linear electric 'motor' consisting of a stator field and an armature placed in that field. This 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 in relation to the magnetic induction.

Rice. 14.1. Primitive linear motor

When voltage is applied to the ends of the 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 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 mechanical energy anchor refers to electrical energy, transmitted to it. Since the armature is located at right angles to the magnetic induction field, the force is determined by the simplified expression 1 1.4

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

Where v – anchor speed. If we apply Kirchhoff's law to a closed circuit, we obtain the following expressions for the current I.

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

Substituting exp. 14.4 in 1 4.3 we get

and substituting expression 14.5 into 14.2 we get

Now let's look at the thermal energy released in the armature. It is determined by vyr. 14.7

Finally, we can express the energy supplied to the armature as

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

We can plot these curves to see how the energy supplied to the armature combines with a range of speeds.(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 have some bearing on coilgun , we will give our variables values ​​that correspond to 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 wire resistance and current will be

R = 0.1

I = 160 A

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

V=16V

All these parameters are necessary to construct 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 frictional force of, say, 2N, so that the reduction in mechanical energy is proportional to the speed of the armature. The value of this friction is deliberately taken higher so that the effect of this is more obvious. The new set of curves is shown in Figure 14.4.

Rice . 14.4. Constant friction performance curves

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

Also worth considering is how force, and therefore acceleration, depends on speed. If we substitute Eq. 14.5 into Eq. 14.1 we get an expression for F in terms of speed v.

Having constructed this dependence, we get the following graph

Rice. 14.5. Dependence of the force acting on the anchor on speed

It is clear that the anchor starts with a maximum accelerating force, which begins to decrease as soon as the anchor 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. Figure 14.6 shows a diagram of the effects of forces on an anchor, from which the resulting force described by the differential equation can be determined.

Rice. 14.6 Diagram of the influence of forces on an anchor

Fm and Fd – magnetic and counteracting forces, respectively. Since the voltage is constant value, we can use equation 14.1 and the resulting force F a , acting on the anchor, will be

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

This is a second order inhomogeneous differential equation with constant coefficients and it can be solved by defining an additional function and a partial integral. Direct line solution method (all mathematics university programs consider differential equations), so I'll just 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 characteristics 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 created by the magnetized projectilecoilgun(this radial component creates the axial force). To do this, it is necessary to integrate the expression obtained by multiplying the current densityDetermination of the volume integral of the radial magnetic flux density usingFEMM

The projectile becomes magnetized when we define for itB- Hcurve andHcvalues ​​inFEMMmaterial properties dialog. ValueswereselectedForstrictcomplianceWithmagnetizediron. FEMMgives a value of 6.74x10 -7 Tm 3 for the volume integral of magnetic flux densityB coil, so usingF= /4 we getB model = 3.0 x10 -2 Tl. This magnetic flux density value may seem very small when 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, incoilgunthe stationary copper surrounds the magnetized part, which moves. 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 projectile, let’s assign it a mass of 12g.

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

Rice. 14.8. Dynamic behavior of a linear motor

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

Rice. 14.9. Characteristics of the dependence of speed on movement

It is important to note here that a relatively long accelerator is required for the armature to begin to reach its maximum speed. ThishasmeaningForconstructionmaximum effectivepracticalaccelerator.

If we zoom in on the curves we can determine what speed will be achieved over a distance equal to length active material in the accelerator gun coil (78 mm).

Rice. 14.10. Increased speed versus displacement curve

These are remarkably close specs to those of an actual manufactured three-stage accelerator, however, this is purely coincidental as there are several significant differences between this model and the actualcoilgun. For example, incoilgunforce is a function of speed and coordinates of movement, 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. Overall efficiency as a function of displacement excluding friction losses

Rice. 14.11. Overall efficiency as a function of displacement taking into account constant friction losses

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

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

Rice. 14.11. InfluenceBon a velocity-displacement gradient

Rice. 14.12. Region of small movement where increasing induction produces greater speed

This set of curves shows an interesting feature of this model, in which a large field inductance in the initial stage gives a large speed in specific point, but as soon as the speed increases, the curves corresponding to the lower inductance catch up with this curve. This explains the following: You have decided that a stronger induction will produce a greater initial acceleration, however, in accordance with the fact that a greater 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 important thing to understand this is that, starting from a dead center, the efficiency of such a motor is very low, especially if the motor is short. Instantaneous efficiency increases as the projectile picks up speed due to the induced voltage reducing the current. This increases efficiency because the energy lost in resistance (obviously heat loss) decreases and the mechanical energy increases (see Fig. 14.3, 14.4), however, since the acceleration also decreases, we obtain progressively greater displacement, so the best efficiency curve will be used.(In short, a linear motor is subject to a step voltage "forcing function" is going to be quite an inefficient machine unless it is very long.)

This model of a primitive motor is useful in that it shows a case of typical poor efficiency.coilgun, namely low level driving induced voltage. The model is simplified and does not take into account the non-linear and inductive elements of a practical circuit, so to enrich the model we need to include these elements in our electrical diagram models. In the next section you will learn the basic differential equations for 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 the Runge Kutta numerical integration algorithm.

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), then it’s 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. Let's 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 value at all points on the sphere. Our rule then states that the field on the surface of a sphere multiplied by 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, its area increases as the square of the radius. The product of the average normal component of the electric field by this area must still be equal to the internal charge, which means that the field must decrease as the square of the distance; This is how a 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, it turns out that it is in general case is not equal to zero (although this is true in a Coulomb field). Instead, the second law holds true 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, limited curve WITH:

The constant c 2 that appears 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 is set so that the usual units of electric current strength arise.

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

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

We can illustrate some laws of electrodynamics with a series of simple experiments that can show us, at least qualitatively, the relationship between the electric and magnetic fields. You become familiar with the first term in equation (1.1) when 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 hanging over a magnetic bar, as shown in Fig. 1.6. When the current is turned on, the wire moves due to the force F=qvXB acting on it. When goes to the wire current, the charges inside it move, that is, they have a speed v, and they are acted upon by the magnetic field of the magnet, as a result of which the wire moves to the side.

When the wire is pushed to the left, you can expect the magnet itself to experience a push to the right. (Otherwise, the 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 rod, the movement of a more sensitive device, say a compass needle, is quite noticeable.

How does current in a wire push a 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, the 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 wand creating a field near the wire IN.

When current flows through a wire, the wire moves 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. For those curves (circles, for example) that lie far from the wire, the length turns out to be greater, so the tangent component B should decrease. You can see that you would expect B to decrease linearly with distance from a long straight wire.

We said that current flowing through a wire creates 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.

This means that 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, which also carries current. 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, could it be that the magnetic field created by a piece of iron is actually the result of currents? Apparently this is true. In our experiments, we can replace the magnetic rod with a coil of wound wire, as shown in Fig. 1.9. When current passes through the coil (as well as through the straight wire above it), exactly the same movement of the conductor is observed as before when there was a magnet instead of the coil. Everything looks as if current were continuously circulating 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 along atomic orbits. This is not the case with iron, but in some materials this is the origin of magnetism. 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, creating a magnetic field in the iron. (We said "something like the rotation of the Earth" because in reality quantum mechanics the question is so deep that it does not fit well enough into classical concepts.) In most substances, some of the electrons rotate in one direction, others in the other, so that magnetism disappears, and in iron (by mysterious reason, which we will talk about later) many electrons rotate so that their axes point in one direction and this serves as a 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. In these equations we're talking about about everyone 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), magnetic charges, similar to the electric charges on the right side of equation (1.6), do not exist. They were never 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 reality such currents exist; The following example illustrates 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 through which flows

There will still be a force acting on the wire.

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

It is charged by a current flowing into one of the plates and flowing out from the other, as shown in Fig. 1.10. Let's draw a curve around one of the wires WITH and stretch a surface onto it (surface S1, which will cross the wire. In accordance with equation (1.9), the circulation of field B along the curve WITH is given by the magnitude of the current in the wire (multiplied by from 2). But what happens if we pull on a curve another surface S 2 in the shape of a cup, the bottom of which is located between the plates of the capacitor and does not touch the wire? No current, of course, passes through such a surface. 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 should remain the same. Indeed, the first term on the right 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 S 2 the circulation of vector B is expressed through the degree of change in the flow of vector E from one plate to another. And it turns out that the change in E is related to the current precisely in such a way that equation (1.9) turns out to be satisfied. Maxwell saw the need for this and was the first to write the complete equation.

Using the device shown in FIG. 1.6, another law of electromagnetism can be demonstrated. Let's disconnect the ends of the hanging wire from the battery and connect 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, and 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 are deflected 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.

Suppose, however, that we left the wire alone and began to move the magnet. We feel that there should be no difference, because relative motion the same thing, and indeed 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 a set of almost very important phenomena happening in electric generators and transformers.

Most remarkable consequence of our equations is that by combining equations (1.7) and (1.9), we can understand why electromagnetic phenomena spread 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 a wire. Then from equation (1.7) it follows that circulation of the electric field should arise. When the electric field begins to gradually increase for circulation to occur, then, according to equation (1.9), magnetic circulation should also arise. But increasing this magnetic field will create a new circulation of the electric field, etc. In this way, the fields propagate through space without the need for charges or currents anywhere other than the source of the fields. This is the way we we see each other! All this is hidden in the electromagnetic field equations.

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All topics in this section:

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 rad wire

Long solenoid
Another example. Let us again consider an infinitely long solenoid with a circumferential current equal to nI per unit length. (We assume that there are n turns of wire per unit length, carrying each

Small loop field; magnetic dipole
Let's use the method vector potential to find the magnetic field of a small current loop. As usual, by “small” we simply mean that we are only interested in fields that are large

Vector circuit potential
We are often interested in the magnetic field created by a chain of wires in which the diameter of the wire is very small compared to the dimensions of the entire system. In such cases we can simplify the equations for magnetic

Biot-Savart's Law
In the course of studying electrostatics, we found that electrical

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

where is some constant (read epsilon-zero). If there are no charges inside the surface, but there are charges outside it (even very close), then the average normal component is still 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. Let's draw a sphere around a point charge. Then the average normal component is exactly equal to the value 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 a sphere multiplied by 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, its area increases as the square of the radius. The product of the average normal component of the electric field by this area must still be equal to the internal charge, which means that the field must decrease as the square of the distance; This is how a 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, it turns out that in the general case it is not equal to zero (although in a Coulomb field this is so). Instead, the second law holds true 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:

(1.8)

And for a surface bounded by a curve:

The constant that appears 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 is set so that the usual units of electric current strength arise.

Equations (1.6) - (1.9), as well as equation (1.1) are all the laws of electrodynamics. As you remember, Newton's laws were very simple to write, but many complex consequences followed from them, so it took a lot of time to study them all. The laws of electromagnetism are incomparably more difficult to write, and we should expect that the consequences from them will be much more complicated, and now we will have to understand them for a very long time.

We can illustrate some laws of electrodynamics with a series of simple experiments that can show us, at least qualitatively, the relationship between the electric and magnetic fields. You become familiar with the first term in equation (1.1) when 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 hanging over a magnetic bar, as shown in Fig. 1.6. When the current is turned on, the wire moves due to the force acting on it. When current flows through a wire, the charges inside it move, that is, they have a speed v, and they are acted upon by the magnetic field of the magnet, as a result of which the wire moves to the side.

When the wire is pushed to the left, you can expect the magnet itself to experience a push to the right. (Otherwise, the 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 rod, the movement of a more sensitive device, say a compass needle, is quite noticeable.

How does current in a wire push a 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 the circulation of the vector; in our case, the field lines are closed around the wire, as shown in Fig. 1.7. It is this field that is responsible for the force acting on the magnet.

Figure 1.6. A magnetic stick that creates a field near a wire.

When current flows through a wire, the wire moves due to the force.

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

Equation (1.9) tells us that for a given amount of current flowing through a wire, the field circulation 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 should decrease. You can see that you should expect a linear decrease with distance from a long straight wire.

We said that current flowing through a wire creates 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. This means that 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, which also carries current. 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.

Figure 1.8. Two wires through which current flows also act on each other with a certain force.

In short, electric currents, like magnets, create magnetic fields. But then what is a magnet? Since magnetic fields are created by moving charges, could it be that the magnetic field created by a piece of iron is actually the result of currents? Apparently this is true. In our experiments, we can replace the magnetic rod with a coil of wound wire, as shown in Fig. 1.9. When current passes through the coil (as well as through the straight wire above it), exactly the same movement of the conductor is observed as before when there was a magnet instead of the coil. Everything looks as if current were continuously circulating 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 along atomic orbits. This is not the case with iron, but in some materials this is the origin of magnetism. In addition to rotating around the atomic nucleus, 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, creating a magnetic field in the iron. (We said “something like the rotation of the Earth,” because in fact in quantum mechanics the question is so deep that it does not fit well enough into classical ideas.) In most substances, some electrons rotate in one direction, others in the other, so magnetism disappears, and in iron (for a mysterious reason which we will talk about later) many electrons rotate so that their axes point in the same direction and this serves as a 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 deal with 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), magnetic charges similar to the electric charges on the right side of equation (1.6) do not exist. They were never discovered.

The first term on the right side of equation (1.9) was discovered theoretically by Maxwell; he is very important. He says that changing electric fields cause 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 reality such currents exist; talks about this next example. Imagine a capacitor made up of two flat plates. It is charged by a current flowing into one of the plates and flowing out from the other, as shown in Fig. 1.10. Let's draw a curve around one of the wires and stretch a surface (surface) onto it that will intersect the wire. In accordance with equation (1.9), the circulation of the field along the curve is given by the magnitude of the current in the wire (multiplied by ). But what will happen if we stretch another surface in the shape of a cup onto the curve, the bottom of which is located between the plates of the capacitor and does not touch the wire? No current, of course, passes through such a surface. But a simple change in the position and shape of an imaginary surface should not change the real magnetic field! The field circulation should remain the same. Indeed, the first term on the right side of equation (1.9) is combined with the second term in such a way that the same effect occurs for both surfaces. For vector circulation, it is expressed in terms of the degree of change in vector flux from one plate to another. And it turns out that the change is related to the current in such a way that equation (1.9) turns out to be fulfilled. Maxwell saw the need for this and was the first to write the complete equation.

Using the device shown in FIG. 1.6, another law of electromagnetism can be demonstrated. Let's disconnect the ends of the hanging wire from the battery and connect them to a galvanometer - a device that records the passage of current through the wire. As soon as you swing a wire in the field of a magnet, 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. Their speed is now directed to the side, because they are deflected along with the wire. This, together with the vertically directed field B of the magnet, results in a force acting on the electrons along the wire, and the electrons are sent to the galvanometer.

Suppose, however, that we left the wire alone and began to move the magnet. We feel that there should be no difference, because the relative motion is the same, and indeed 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), we can understand why electromagnetic phenomena extend to 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 a wire. Then from equation (1.7) it follows that circulation of the electric field should arise. When the electric field begins to gradually increase for circulation to occur, then, according to equation (1.9), magnetic circulation should also arise. But the increase in this magnetic field will create a new circulation of the electric field, etc. In this way, the fields propagate through space without the need for charges or currents anywhere other than the source of the fields. This is the way we see each other! All this is hidden in the electromagnetic field equations.

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Electrical and magnetic phenomena have been known to mankind since ancient times. The very concept of “electrical phenomena” dates back to Ancient Greece(remember: two pieces of amber (“electron”), rubbed with a cloth, repel each other, attract small items...). Subsequently, it was found that there are two types of electricity: positive and negative.

As for magnetism, the properties of some bodies to attract other bodies were known in ancient times, they were called magnets. The property of a free magnet to be established in the “North-South” direction already in the 2nd century. BC used in Ancient China while traveling. The first experimental study of a magnet in Europe was carried out in France in the 13th century. As a result, it was established that the magnet has two poles. In 1600, Gilbert put forward the hypothesis that the Earth is a large magnet: this is the basis for the possibility of determining direction using a compass.

The 18th century, marked by the formation of MKM, actually marked the beginning and systematic research electrical phenomena. So it was established that like charges repel, and the simplest device appeared - an electroscope. In the middle of the 18th century. was installed electrical nature lightning ( research by B. Franklin, M. Lomonosov, G. Richman, and Franklin’s merits should be especially noted: he is the inventor of the lightning rod; It is believed that it was Franklin who proposed the notation “+” and “–” for charges).

In 1759, the English naturalist R. Simmer concluded that in the normal state any body contains an equal number of opposite charges that mutually neutralize each other. During electrification, their redistribution occurs.

At the end of the 19th and beginning of the 20th century, it was experimentally established that an electric charge consists of an integer elementary charges e=1.6×10-19 Cl. This is the smallest charge existing in nature. In 1897, J. Thomson discovered the smallest stable particle, which is the carrier of an elementary negative charge(electron having mass moe=9.1×10-31). Thus, the electric charge is discrete, i.e. consisting of separate elementary portions q=± ne, where n is an integer.

As a result of numerous studies of electrical phenomena undertaken in the 18th and 19th centuries. A number of important laws were obtained.

Law of conservation of electric charge: electrically closed system the sum of charges is a constant value. (That is, electric charges can arise and disappear, but at the same time an equal number of elementary charges of opposite signs necessarily appear and disappear). The amount of charge does not depend on its speed.

The law of interaction of point charges, or Coulomb's law:

Where e is relative permittivity environment (in vacuum e = 1). Coulomb forces are significant up to distances of the order of 10-15m (lower limit). At shorter distances they begin to act nuclear forces(so-called strong interaction). Regarding upper limit, then he tends to:.

The study of the interaction of charges, carried out in the 19th century. It is also remarkable that with him it entered into science concept of field. This began in the works of M. Faraday. Field stationary charges called electrostatic. Electric charge, being in space, distorts its properties, i.e. creates a field. Power characteristics electrostatic field is its tension. The electrostatic field is potential. His energy characteristics serves as potential j.

Oersted's discovery. The nature of magnetism remained unclear until the end of the 19th century, and electrical and magnetic phenomena were considered independently of each other, until in 1820 the Danish physicist H. Oersted discovered the magnetic field of a current-carrying conductor. This is how the connection between electricity and magnetism was established. The strength characteristic of a magnetic field is intensity. Unlike open electric field lines power lines magnetic field are closed, i.e. it is vortex.

Electrodynamics. During September 1820 French physicist, chemist and mathematician A.M. Ampere is developing a new branch of the science of electricity - electrodynamics.

Ohm's, Joule-Lenz's laws: the most important discoveries in the field of electricity, the law discovered by G. Ohm (1826) I=U/R and for a closed circuit I= EMF/(R+r), as well as the Joule-Lenz law for the amount of heat released when current passes through a stationary conductor during time t: Q = IUT.

Works by M. Faraday. The research of the English physicist M. Faraday (1791-1867) gave a certain completeness to the study of electromagnetism. Knowing about Oersted's discovery and sharing the idea of ​​​​the relationship between the phenomena of electricity and magnetism, Faraday in 1821 set the task of “converting magnetism into electricity.” In 10 years experimental work he discovered the law electromagnetic induction. (The essence of the law: a changing magnetic field leads to the emergence induced emf EMFi = k×DFm/Dt, where DFm/Dt is the rate of change of magnetic flux through the surface stretched over the contour). From 1831 to 1855 published in series main work Faraday " Experimental studies on electricity."

While working on the study of electromagnetic induction, Faraday came to the conclusion about the existence electromagnetic waves. Later, in 1831, he expressed the idea of ​​the electromagnetic nature of light.

One of the first to appreciate Faraday's work and his discoveries was D. Maxwell, who developed Faraday's ideas by developing in 1865 the theory of the electromagnetic field, which significantly expanded the views of physicists on matter and led to the creation of the electromagnetic picture of the world (EMPW).

Lecture outline

1. Electrostatics. Brief overview.

2. Magnetic interaction 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. Magnetic field of a moving charge.

1. Electrostatics. Brief overview.

Let's preface the study of magnetostatics brief overview basic principles of electrostatics. Such an introduction seems appropriate, because when creating the theory of electromagnetism, we used methodological techniques, which we have already encountered in electrostatics. That’s why it’s worth remembering them.

1) Basic experienced law 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- not a clever invention of physicists, but an objective reality of nature.

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

,. (E2)

3) Combining the definition of intensity (E2) and Coulomb’s law (E1), we find the strength of the field 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 we can, perhaps, 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 Ampere in 1820.

In Fig. 8.1. A diagram of one of his experimental setups is shown. Here, the rectangular frame 1 can be easily rotated around a vertical axis. Reliable electrical contact when turning the frame was ensured 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 Ampere measured and analyzed, believing that the forces of interaction between the remote edges of the frames could be neglected.

Rice. 8.1.

Ampere experimentally established that parallel currents are 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 acting per unit length of a conductor 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 the interaction of two parallel currents is used in the SI system to define the basic electrical unit - the unit of current 1 ampere.

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

Having thus determined the unit of current, 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
.

Very short time the nature of the force interaction of electric currents remained unclear. In the same 1820, the Danish physicist Oersted discovered the influence of electric current on the 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, positioning itself perpendicular to the conductor with the current.

Rice. 8.3.

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

In magnetostatics, as in electrostatics, we came to the field theory of the interaction of currents, to the concept of short-range interaction.