Shift current. To summarize equations electronic magnetic field In vacuum to the field variables, it is necessary to change only one of the previously written equations (see Section 3.4, 3.12); Three equations are faithful in the general case. However, the law full current For a magnetic field, in the case of variable fields and currents turns out to be incorrect. In accordance with this law, the current must be the same for any two surfaces stretched on the contour; If the charge in the amount between the selected surfaces changes, then this statement enters into a contradiction with the law of saving charge. For example, when charging a capacitor (Fig. 45), the current through one of the specified surfaces is equal to another (passing between the plates) - zero. To remove the specified contradiction, Maxwell introduced into this offset current equation, proportional to the speed of change electric field:
The history of the development of the Maxwell equations and the theory of relativity. This highly-sensitive designation would directly inspire fundamental physics. Maxwell 2 equations in equations, Maxwell equations were adopted to serve only for expressing electromagnetism in an inertial luminous ether system. The experiment Edward Morley and Albert Abraham Maykelson gave a zero result for a hypothesis about changing the speed of light due to the hypothetical movement of the Earth through the air.
However, alternative explanations were looking for other Lorentz. This led to the theory of the special theory of the relativity of Albert Einstein, which postulated the absence of absolute reference and invariance of the Maxwell equations in all references. The electromagnetic field equations have a close connection with special theory Relativity: The magnetic field equations can be obtained from the reasons of the electrical field equations in relativistic transformations at low speeds.
In the dielectric medium, the expression for the bias current takes the form:
The first term is the density of the bias current in vacuo, the second is a real current due to the movement of the associated charges when the polarity changes. The shift current through the surface is where F is the flow of the vector through the surface. The introduction of the offset current removes the contradiction with the law of saving the charge. For example, when charging a flat capacitor, the displacement current through the surface passing between the plates, equal to the current By supply wires.
This strategy for using large measurements to combine different forces is an area of \u200b\u200binterest to explore the physics of particles. Fat variables in equations are vector or vector fields, integrals - surface integrals on a "closed" surface, integrals - surface integrals on an open surface, and integrals - linear integrals via a closed path.
The Law of Gauss Magnetism: Faraday Induction Act: Ampere Maxwell Ampere Expansion. The volumetric density of the electric charge, not counting the dipoles of the associated charges in the material, is surface density Magnetic flux, also called magnetic induction.
The system of Maxwell equations in vacuo. After administration of the offset current, the Maxwell equations system in differential form takes the form:
The system of Maxwell equations in integral form:
Electrical offset field or surface density of the electric field. Electric field strength, magnetic field strength - surface density electric current. It is a gradient operator, which in Cartesian coordinates can be recorded as.
Drying vector field, this is a field of rotation vector. The second equation determines the non-existence of magnetic monopoles. The force acting on the charged particle with an electric field, and the magnetic field is determined by the Lorentz power equation. In which there is a particle charge and particle speed.
We also give the recording of Maxwell equations in differential form in the SGS system:
Charge density and current are associated with relation
expressing the law of preservation of the charge (this equation is a consequence of the Maxwell equations).
Maxwell Equations in Wednesday View: Differential Shape Integral Form
It is important to note that the Maxwell equations are usually applicable to "macroscopic means" of fields that can vary greatly in a microscopic scale in the vicinity of individual atoms. Only in this middle sense can be determined by the values, such as solvability and permeability of the material below.
Where: ε is a dielectric constant or electrical solvability. μ - magnetic permeability. In a homogeneous medium, ε and μ are independent of the position by constants and therefore can be replaced by spatial derivatives. In a more general case, ε and μ may be second-order tensors describing birefringent materials.
and serve to determine four quantities. To the Maxwell equations, in the medium it is necessary to add material equations of communication between, characterizing the electrical and magnetic properties of the medium. For isotropic linear media, these equations look:
From the Maxwell equations, it is possible to obtain boundary conditions for (see Section 3.6, 3.13).
In vacuo without loads or chains. Vacuum is a linear, homogeneous and isotropic medium, and its electrical constants are denoted by ε0 and μ0. If there are no currents or electrical charges, then the Maxwell equations are obtained in vacuo. These equations have a simple solution in terms of sinusoidal flat progressive waves with the directions of orthogonal magnetic and electric fields to each other and the direction of displacement and with two fields in the phase.
But: which allows to obtain an electromagnetic wave equation. Where does the speed of the electromagnetic wave come from. Maxwell realized that this value C is just the speed of light in vacuum, and came to the conclusion that the light is the form of electromagnetic radiation.
The law of energy conservation for the electromagnetic field.
From the Maxwell equations, you can withdraw the following equation for any volume V, limited to the surface
The first term describes the change in the energy of the electromagnetic field in the volume under consideration. It can be seen that in general, for the energy density of the electromagnetic field, the formulas obtained earlier for permanent electrical and magnetic fields are faithful. The second member is the work of the field over the particles in the volume under consideration. Finally, the third term describes the flow of electromagnetic energy through the limiting volume of the closed surface. The energy flow density at this point of space (Pointing vector) is determined by vectors E and in the same point:
Density of charge I. electric field. Equivalent integral form, also known as Gauss law, is such. According to the divergence theorem: and according to the law of Gauss. In this way. Where is the free density of the electric charge, not counting the dipoles of related charges in the material, and this field electric displacement. This equation corresponds to the Culon law for stationary charges in vacuo.
In the linear material, it is directly associated with an electric field depending on the material constant, called the permissiveness. Any material can be considered as linear if the electric field is not extremely intense. Where, again, this is an electric field, this is a complete charge density, and it the dielectric constant in vacuum. It can also be written in the form: where is the relative dielectric permeability of the material or its dielectric constant.
The last expression is valid for the density of the electromagnetic energy flow in the substance. Energy density in the medium has the form:
Example 1. Consider charging a flat condenser with round plates located at a distance. The rate of energy change in the cylinder with a radius (less than the size of the plates) is equal
Compare with the Poisson equation. The magnetic field structure is a magnetic flux density, also called magnetic induction. The area of \u200b\u200bthe differential square with the surface is normal, directed, which determines its direction.
Note. Like the integral form of the electric field, this equation works only if the integral is calculated on a closed surface. This equation is associated with the magnetic field structure, since it claims that at the same time a given volume element, the net magnitude of the vector components indicating the surface should be equal to the magnitude of the vector component indicating inside. Structurally, this means that the magnetic field lines.
Magnetic field tension will find from the second equation of Maxwell: (the offset current is right). We obtain that the speed of the inflow of energy through the side surface of the cylinder: equal to the rate of energy change in the amount.
Relativistic properties of fields. When moving from one inertial reference system to another, both sources of the electromagnetic field (charge and current density) are changed and the fields themselves, but the Maxwell equations retain their own appearance. The easiest way of conversion formulas for sources are the density of the moving charge). If you designate the charge density in ISO, in which it is taking into account the reduction of longitudinal sizes (see Section 1.11) we get
Maxwell 6 equations should be closed. Another way to declare that field lines cannot come from another place; Trying to return the lines back to the starting position. Therefore, it is a mathematical formulation of the hypothesis that there are no magnetic monopoles.
Variables magnetic and electric fields. Equivalent integral form: Using the Stokes theorem, we have. Electromotive force equal to the value of this integral. This law corresponds to the law of electromagnetic induction of Faraday. Pay attention to the negative sign; This is necessary to save energy. It is so important that he has his name, Lenza law.
Comparing with an empty-pulse-vector, we see that the substrate is formed, i.e. Converted to each other in the same way as the Lorentz transform formulas. Knowing how sources of field are transformed, you can find formulas for conversion E, V. They look like this:
This equation relates to electrical and magnetic fields, but it also has several practical applications. This equation describes how electric motors and electrical generators work. In particular, it demonstrates that "voltage" can be caused by a change in the magnetic flux passing through the specified area in time, just as the coil rotates evenly through a fixed magnetic field.
In the engine or generator, the fixed excitation is provided by a field circuit, and the alternating voltage is measured by an anchor circuit. Note: Maxwell equations are applicable to the right coordinate system. Application of them unchanged to the left coordinate system would mean the exchange of polarity of magnetic fields.
Here - the speed of the reference system to the system to, the conversion is recorded for the field components parallel and perpendicular to the invariants of these transformations are scalar quantities
With the field conversion formula, the following simplified view takes:
Example 2. Magnetic field of an unrelativistic particle. Consider a particle that moves relative to ISO with a constant nonrelativistic velocity V. In an ISO associated with a moving particle, there is only an electric field for switching to ISO to write formulas
Maxwell Equations 7 The source of the magnetic field. Using the Stokes theorem, we have. AMPER Act: Maxwell's contribution. Student difficulties in the study of Gaussian law at the level of general physics in the light of the theory of Menantal models of Johnson-Lardda. Students "Difficulties in the study of the Gauss Law" at the introductory level of college in the light of the theory of mental models Johnson-Lard.
In this paper, we present an analysis of the answers of 74 university students to questions regarding the Gaussian law on electricity in the discipline of general physics. These difficulties are analyzed in the theoretical framework of Johnson-Lardd's mental models. The reason that the generator of these difficulties would have concluded that the students would not be able to build mental models and appliances schemes that would attach meaning to the concepts involved and, especially, the law itself. Probably the instructions received would not be sufficient to facilitate the construction of such models.
transformations Considering that in the nonrelativistic limit of length lengths do not change, we obtain (for the moment when the particles go to the beginning of the coordinates):
In the conclusion of these formulas, equality was used
Example 3. Polarization of the dielectric when moving in a magnetic field. When the dielectric movement with nonrelativistic speed perpendicular to the lines of the magnetic field induction lines occurs its polarization. In an Iso associated with a dielectric, there is a transverse electric field. The nature of the polarization of the dielectric depends on its shape.
Keywords: physical learning, Gauss law, mental models. This article presents an analysis of the answers of 74 college students to questions about the law of Gauss for electricity in the entrance course of physics. These difficulties were considered in the light of the mental models of Johnson-Lard. The main reason for such difficulties will be the fact that the disciples could not build mental models and, moreover, the schemes of assimilation that could give some values \u200b\u200bto the law and the concepts involved. Perhaps they could not help them in the construction of these models and schemes.
Example 4. Electrical field of relativistic particle. Consider a particle that moves relative to ISO with a constant relativistic velocity V. In ISO to a moving particle associated with a moving particle, there is only an electric field for switching to ISO to use the conversion formulas (92) with writing the answer for the moment when the particle is in ISO It passes through the origin, for a point lying in the plane in the transition from coordinates to the coordinates it is necessary to take into account that (the coordinates of the point are measured in to the passage of the particle through the origin of the coordinates). As a result, we get
As a rule, electric charge begins and electrical powerwhich soon moves to the concept of an electric field. Further, the Law of Gauss is introduced as the general law of electromagnetism, one of the Maxwell equations, which in electrostatic tasks is equivalent to the Culon law, but which greatly simplifies the calculations when the symmetry of the problem is large. Well, despite the "advantages" of the Gauss Law in solving some problems, it seems to be the beginning of the difficulties of students in the study of electromagnetism.
Anyone who taught the Gaussian law on electricity in general physics disciplines probably created the impression that the disciples did not understand him physical meaningHe hardly applied him to certain problems and classified it as one of the formulas of physics.
It can be seen that the vector E collinearin vector however, on the same distance from charge the field at a point located on the line of its movement, less than at a point located on the perpendicular to the speed. The magnetic field at the same point is determined by the expression:
Note that the electrical field considered is not potential.
Having received such an impression from many times when we tried to teach Gauss's law, we decided to more carefully examine the set of students' answers to try to classify their difficulties, and then interpret them in the light of theoretical foundations. We will begin our analysis of our analysis, describing how the Gauss law is presented in the textbook used at a time, since theoretical and problem classes were based on this text, and students studied the subject. In continuity, we will present questions, the answers to which were analyzed, and the categories of difficulties.
Four equations corresponding to our (modified) allegations are called maxwell equations in integrated form.
Let's write them all near:
To obtain the Maxwell equations in the medium, it is necessary to replace:
that is, indicate the connection (the so-called "material" equations) between tensions and inductions: and and supplement the system by the Ohma law equation
Finally, we will try to interpret such difficulties from the point of view of the theory of mental models. Such work will be mentioned in the final part of this article. Initially, we are talking about the Law of Gauss as a new way to formulate the law of Culon. In addition, from the very beginning it should be noted that this law should be used when "symmetry" approximately "high".
Gaussian surface is introduced as the central part of the Gauss law. The main figure of the Gauss law is a hypothetical closed surface, called a Gaussian surface. Then we will discuss Gaussian form, stressing that it should be an adequate symmetry of a problem that is solved and that it should always be closed, which often leads to the formation of a spherical or cylindrical surface or some other symmetric form.
Note that the simplest ratios can not always be used. The situation is noticeably more complicated in the presence of such substances as ferroelectrics, piezoelectrics, ferromagnets, anisotropic substances, and the like. Here our goal is to show how a complete system of equations is formed, allowing (taking into account the initial and boundary conditions, of course) to calculate the electromagnetic field.
From equations in the integral form, using the vector analysis theorems, go to the differential equations that bind the values \u200b\u200bof the fields and their spatial and temporary derivatives with the values \u200b\u200bof charge and current density values. We will not use these equations, but still we will give them at least as part of a joke published in one of the journals in the days of the anniversary of Maxwell:
"And God said:
And the light became. "
Incomprehensible icons div (read " divergence") I. rot. (read " rotor") - These are special differentiation operations performed above vector fields. Divergence - in Latin "Discrepancy". This operation describes the configuration of lines of type "hedgehog", divergent from the points where there are electric charges. The word "rotor" does not need translated, it is clearly associated with rotation. This operation describes the vortex fields (ring-shaped - closed power lines) Around their sources - currents or other fields varying in time.
Four integral equations and four differential are equivalent. Maxwell showed that all phenomena of electromagnetism can be fully described by these four equations that are generalized by experimental facts.
The last joke mentioned light. Indeed, light is the electromagnetic radiation of a specific frequency range. The prediction of electromagnetic waves has become one of the greatest achievements of Maxwell's theory. Let us imagine that there are no charges and currents. Let's look at the Maxwell equations in differential form. It can be seen that if the fields are not static, but depend on the time, then there is a vortex electrical and magnetic fields (the corresponding rotors are different from zero). The spread of fields without charges and currents is electromagnetic waves. And you can coal in the hint of the speed of their distribution: there is a combination E 0 m 0, through which it can be expressed vacuum light speed(see (6.3))
But about it later, in the next part of our course.
In the conclusion of this part, we quote the words of Gersa about the Maxwell equations:
"It is difficult to get rid of the feeling that these mathematical formulas live independent life and possess their own intelligence that they are wiser than we ourselves, wiser even than their discovers, and that we remove from them more than it was initially laid in them."
An example of using Maxwell equations
Determine the magnitude of the magnetic field in the condenser's gap as a function R of distances from the axis of symmetry (Fig. 9.13)
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Fig. 9.13. Round Plates Condenser in the process of charging
Decision
We write equation (9.13) for the contour shown in Fig. 9.3 stroked line. Integrating, get
Obviously, the magnetic field is not zero only due to the presence of a changing electric field. In turn, the change in the electric field is due to an increase in the charge on the condenser plates. This connection will be obtained from relations
Finally find
