- •1. Electrostatic field. Coulomb’s law. Gauss law (Электростатическое поле. Закон Кулона. Закон Гаусса)
- •Variables and units
- •Coulomb’s Law. (ЗаконКулона)
- •Electric Field Strength e and Displacement Field d. (Напряжённостьисмещениеэлектрическогополя)
- •Gauss’ Law. (ЗаконГаусса)
- •2. Poisson’s and Laplace’s equations for the potential of electric field (Уравнения Пуассона и Лапласа для потенциала электрического поля) Electric Potential. (Электрический потенциал)
- •Poisson’s and Laplace’ s equations. (Уравнения Пуассона и Лапласа)
- •3. Electrostatic Energy (Электростатическая энергия) Electrostatic Energy (Электростатическаяэнергия)
- •Virtual experiment. (Эксперимент по нахождению энергии системы)
- •Consequences (Следствия)
- •4. Power and Joule’s Law (Энергия и закон Джоуля-Ленца)
- •5. Continuity Equation (Уравнения непрерывности) ContinuityEquation (Уравнение непрерывности)
- •Image method for the flat boundary between magnetic media (Метод изображений для плоской границы между магнитными носителями)
- •8. Static magnetic field. Biot–Savart’s Law. Ampere’s Law (Статическое магнитное поле. Закон Био–Савара. Закон Ампера)
- •Variables and units (Переменные и единицы измерения)
- •Main Relations (Основные соотношения)
- •Magnetic flux density (Индукция магнитного поля)
- •Biot-Savart’s law (Закон Био-Савара)
- •Ampere’s law (Закон полного тока)
- •The cut in the space (Разрез в пространстве)
- •Laplace equation for the scalar magnetic potential (Уравнение Лапласа для скалярного магнитного потенциала)
- •10. Vector magnetic potential. Inductance (Векторный магнитный потенциал. Индуктивность)
- •Vector magnetic potential (Векторный магнитный потенциал)
- •Magnetic flux (Магнитный поток)
- •Differential equation for the vector magnetic potential (Дифференциальное уравнение для векторного магнитного потенциала)
- •Gauging of the vector magnetic potential (Калибровка векторного магнитного потенциала)
- •Integral presentation of the vector magnetic potential (Интегральное представление векторного потенциала)
- •Inductance (Индуктивность)
- •Mutual inductance (Взаимная индуктивность)
- •Inductance of thin contours (Индуктивность тонких контуров)
- •12. Internal inductance of a thin conductor (Внутренняя индуктивность тонкого проводника) Flux linkage of a thin current layer (Потокосцепление тонкого слоя с током)
- •Internal inductance of a thin conductor (Внутренняя индуктивность тонкого проводника)
- •13. Inductance of a two wire transmission line (Индуктивность двухпроводной линии).
- •14. Variable separation method in a cylindrical coordinate system (Метод разделения переменных в цилиндрической системе координат). Application of Laplace’s equation (Применение уравнения Лапласа).
- •Angular function (Угловая функция)
- •Radial function (Радиальная функция)
- •General solution of the Laplace’s equation in a cylindrical coordinate system (Общее решение уравнения Лапласа в цилиндрической системе координат)
- •15. The Faraday’s law (Закон электромагнитной индукции).
- •Lenz’s Law (правило Ленца)
- •Induction by a temporal change of b (Индукция за счёт временного изменения b)
- •16. Induction through the motion of a conductor (Индукция за счет движения проводника).
- •17. Induction by simultaneous temporal change of b and motion of the conductor (Индукция одновременным изменением b во времени и движением проводника).
- •18. Unipolar generator (Униполярный генератор).
- •19. Hering’s paradox (Парадокс Геринга)
- •20. Diffusion of magnetic fields into conductors (Распространение электромагнитного поля в проводнике)
- •21. Periodic electromagnetic fields in conductors. (Периодическое электромагнитное поле в проводниках)
- •Penetration of the electromagnetic field into a conductor. (Проникновение электромагнитного поля в проводник)
- •The skin effect. (Скин-эффект)
- •22. Poynting theorem. (Теорема Пойнтинга) Electromagnetic Field Energy. (Энергия электромагнитного поля)
- •The rate of decrease of the electromagnetic field energy in a closed volume. (Скорость уменьшения энергии электромагнитного поля в замкнутом объёме)
- •Transmission of energy in a dc line (Передача энергии в линиях постоянного тока)
- •The field picture near the wires with current (Картина поля вблизи провода с током)
- •25. Energy flows in static electric and magnetic fields (Поток энергии в статических электрических и магнитных полях).
- •26. The reduced magnetic potential (Редуцированный магнитный потенциал). Reduced scalar magnetic potential (Редуцированный скалярный магнитный потенциал)
- •Combination of scalar magnetic potential and reduced magnetic potential (Комбинация скалярного магнитного потенциала и редуцированного магнитного потенциала)
- •27. Classification of numerical methods of the electromagnetic field modeling (Классификация численных методов моделирования электромагнитного поля).
- •Classification of the problems (Классификация проблем)
- •Classification of the methods (Классификация методов)
- •28. Method of moments
- •Discretization of the problem domain (Дискретизация проблемной области)
- •29. Basic principles of the finite element method.
- •30. Finite functions (Ограниченная функция – отлична от нуля только в пределах треугольника)
- •Simplex coordinates
- •Approximation of functions inside triangles (Аппроксимация функций внутри треугольника)
- •Approximation of the equation (Аппроксимация уравнения)
- •31. Weighted residual method (метод взвешенных невязок)
- •32. Weak formulation of the electromagnetic field modeling problem
- •33. Boundary conditions in electric and magnetic fields
- •1) First type boundary conditions
- •34. Main equations of electromagnetic field in integral form.
- •35. Main equations of electromagnetic field in differential form.
- •36. Electric field of a point charge (Электрическое поле точечного заряда)
- •37. Electric field of a uniformly charged sphere (Электрическое поле равномерно заряженной сферы)
- •38. Flat capacitor. Field. Surface charge. Capacity. (Плоский конденсатор. Поле. Поверхностный заряд. Вместимость.)
- •39.2 Inductance of a cylindrical coil with the rectangular cross section(Индуктивность цилиндрической катушки прямоугольного сечения).
- •4 0.1 Electric field induced by charged line placed above conducting surface (Электрическое поле, создаваемое заряженной линией, помещенной над проводящей поверхностью).
- •4 0.2. Magnetic field induced by the line with a current placed above a ferromagnetic surface with infinitely high magnetic permeability
17. Induction by simultaneous temporal change of b and motion of the conductor (Индукция одновременным изменением b во времени и движением проводника).
Let's look what will happen if simultaneously the configuration of the electromagnetic field changed in time and the magnetic field itself also changed in time. In such a case, both effects should be taken into account, both of them will induce parts of electric field intensity. If now we shall consider again the integral of Edl product around the closed loop, then we can split this relation into two parts. The first of them is what is going on with the frame, contour if only magnetic field changed in time. Another part of the Faraday's Law is what is going on with the contour if magnetic field is stable, static but the contour changes it's configuration. And finally we shall get this relation:
This relation describes, corresponds to the Faraday's Law in the case of moving conductor and the same relation is valid in the case when both coil configuration and magnetic flux density depends on time. In both cases this full relation is correct and should be used.
Let's look at some properties, general properties of the electromagnetic field when this universal (both flux density and coil configuration depend on time) case is consider. Using the vector magnetic potential:
(The Gauss theorem for curl of vector)
Or
The last expression is true for any surface so:
Conclusion: Curl of this parenthesis equal to zero. Why it's important? We have talked already about the vectors which have zero curl. So if the curl of function equal to zero, then we can express it as gradient of some function. Let's do it here also.
Introducing a suitable scalar function , we may write: that is a general expression which may be used in the case of static field, dynamic field, fields which changes in time, static field with the moving conductor.
H ere we can express separately vector E: the field intensity E here is the one which an observer would "see" when moving with the conductor.
An observer at rest then sees the field:
18. Unipolar generator (Униполярный генератор).
Here we have a rotating wheel, conducting wheel and external magnetic flux density, which is parallel to the axis of this wheel. This wheel is rotating with some angular velocity and there are brush contacts. One of this brush contacts is attached to the axis, another one is attached to the circular part of the wheel. This contour is vertical, it is not crossed by the magnetic field at all.
The result of this closed circuit is an induced EMF.
How the voltage may be induced in this contour? We have no changing magnetic flux. But the experiment show us there is the voltage.
The flux passing through the contour is constant! Ф=0
Velocity is a function of the distance to the center:
Electric field intensity:
So, the field really will be induced, but what to do with the flux? It's always equal to zero, theory tells us that in the universal case the induced electromotive force is equal to full derivative dФ/dt with the minus sign, not applicable for this case.
T he second impression we should look at this system even in more details. The contour is more complex that we have seen in the beginning. The current goes along this line, but then really it can't come through this line, because this is the moving object. And current comes really along the radius, but radius moves by itself. Finally, current will move along such curve line. Nevertheless, let's try to analyze the magnetic flux which is crosses this triangle. The area of this triangle is
T he magnetic flux in this case is:
The induced EMF is:
However, this explanation is not very persuasive. What is always true: Using the flux in this certain situation makes no sense.