Lecture 1. - Revision history

Marcsa: /* Boundary Conditions */

2026-02-26T20:20:54Z

‎Boundary Conditions

Marcsa: /* Interface Conditions */

2020-09-10T08:13:12Z

‎Interface Conditions

Marcsa at 18:51, 28 January 2020

2020-01-28T18:51:10Z

Marcsa: /* Interface Conditions */

2019-10-10T07:32:41Z

‎Interface Conditions

Marcsa: /* Constitutive Relations */

2019-10-01T15:46:11Z

‎Constitutive Relations

Marcsa: /* Interface and Boundary ConditionsM. Kuczmann - The Finite Element Method in Electromagnetics, Presentation, University of Győr, 2016. */

2019-04-08T18:08:47Z

‎Interface and Boundary ConditionsM. Kuczmann - The Finite Element Method in Electromagnetics, Presentation, University of Győr, 2016.

Marcsa: /* Interface and Boundary Conditions */

2019-04-08T18:08:25Z

‎Interface and Boundary Conditions

Marcsa at 18:07, 8 April 2019

2019-04-08T18:07:41Z

Marcsa: /* Basic Principle of Electromagnetic Theory */

2019-03-27T19:13:18Z

‎Basic Principle of Electromagnetic Theory

Marcsa at 19:12, 27 March 2019

2019-03-27T19:12:38Z

@@ Line 185: / Line 185: @@
 ::<math>\frac{\partial V(\vec{r})}{\partial n}=\nabla V \cdot \vec{n} = 0</math>,
-on outer surface of problem, that is directional derivative of <math>V</math> along the outward normal <math>\vec{n}</math> to the boundary equals zero. This condition implies that <math>\partial H/\partial n = 0</math> on a perfect electric conductor, since normal derivative of <math>H</math> is proportional to tangential electric field. The boundary conditions are called homogeneous boundary conditions bacause the right-hand side of the equation is zero. The corresponding ''inhomogeneous boundary conditions'' are obtained when the right-hand side is not zero, for example,
+on outer surface of problem, that is directional derivative of <math>V</math> along the outward normal <math>\vec{n}</math> to the boundary equals zero. This condition implies that <math>\partial H/\partial n = 0</math> on a perfect electric conductor, since normal derivative of <math>H</math> is proportional to tangential electric field. The boundary conditions are called homogeneous boundary conditions because the right-hand side of the equation is zero. The corresponding ''inhomogeneous boundary conditions'' are obtained when the right-hand side is not zero, for example,
 ::<math>V(\vec{r})=\text{constant}</math>;

@@ Line 138: / Line 138: @@
 The interface conditions between two materials, as shown in figure, prescribe continutity for the tangential component of the electric field intensity,
-::<math>\vec{n}\times\left(\vec{E}_{2} - \vec{E}_{1} = \vec{0}\right)</math>.
+::<math>\vec{n}\times\left(\vec{E}_{2} - \vec{E}_{1}\right) = \vec{0}</math>.
 The surface current density <math>\vec{K}</math> relates to the tangential component of the magnetic field intensity vector,

@@ Line 6: / Line 6: @@
 | width=50% |
 '''Instructor'''
-* Dániel Marcsa (lecturer)
+* [http://wiki.maxwell.sze.hu/index.php/Marcsa Dániel Marcsa] (lecturer)
 * Lectures: Monday, 14:50 - 16:25 (D201), 16:30 - 17:15 (D105)
 * Office hours: by request

@@ Line 136: / Line 136: @@
 ==== Interface Conditions ====
 [[File:Interface.PNG|400px|thumb|left|alt=For the behavior of electric and magnetic field quantities along the interface.|For the behavior of electric and magnetic field quantities along the interface.]]
-The interface conditions between two media, as shown in figure, prescribe continutity for the tangential component of the electric field intensity,
+The interface conditions between two materials, as shown in figure, prescribe continutity for the tangential component of the electric field intensity,
 ::<math>\vec{n}\times\left(\vec{E}_{2} - \vec{E}_{1} = \vec{0}\right)</math>.

@@ Line 112: / Line 112: @@
 This can be written in another way
-::<math>\vec{B}=\vec{B}(\vec{H})</math>,
+::<math>\vec{B}=\mathfrak{B}(\vec{H})</math>,
-::<math>\vec{J}=\vec{J}(\vec{E})</math>,
+::<math>\vec{J}=\mathfrak{J}(\vec{E})</math>,
-::<math>\vec{D}=\vec{D}(\vec{E})</math>,
+::<math>\vec{D}=\mathfrak{D}(\vec{E})</math>,
-where <math>\vec{B}(\cdot)</math>, <math>\vec{J}(\cdot)</math> and <math>\vec{D}(\cdot)</math> are operators.
+where <math>\mathfrak{B}(\cdot)</math>, <math>\mathfrak{J}(\cdot)</math> and <math>\mathfrak{D}(\cdot)</math> are operators.
 If the material properties are independent of space <math>\vec{r}</math>, they are ''homogenious'', otherwise they are ''inhomogeneous'', <math>\mu=\mu(\vec{r})</math>, <math>\sigma=\sigma(\vec{r})</math>, <math>\varepsilon=\varepsilon(\vec{r})</math>. Constitutive relations may depend on the frequency of excitation as well, <math>\mu=\mu(f)</math>, <math>\sigma=\sigma(f)</math>, <math>\varepsilon=\varepsilon(f)</math>. When the constitutive parameters depend on the directions of the applied field, the materials are ''anisotropic'', otherwise they are ''isotropic''. In the anisotropic case the permeability, the conductivity and the permittivity are tensors, <math>\vec{B}=[\mu]\vec{H}</math>, <math>\vec{J}=[\sigma]\vec{E}</math>, <math>\vec{D}=[\varepsilon]\vec{E}</math>, for example

@@ Line 132: / Line 132: @@
 ::<math>\vec{B}=\mathfrak{B}\{\vec{H},\vec{r},f\}</math>.
-=== Interface and Boundary Conditions<ref>[https://drive.google.com/file/d/1DmkzXA8BZzlSZTPkjWbI093_mion6ETw/view?usp=sharing M. Kuczmann - The Finite Element Method in Electromagnetics, Presentation, University of Győr, 2016.]</ref> ===
+=== Interface and Boundary Conditions ===
 Maxwell's equations along with the constitutive relations my be used to obtain general solution for the electromagnetic problems. To obtain unique solutions, we must enforce the boundary conditions at the periphery of the problem. Additionally, in a mixed media device (<math>\mu_{1}; \mu_{2}; \varepsilon_{1}; \varepsilon_{2}; \sigma_{1}; \sigma_{2}</math>), continuity conditions at the interface of two media should be satisfied in order to ensure continuity of fields across the interface.
 ==== Interface Conditions ====

@@ Line 20: / Line 20: @@
 === [https://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell's Equations] ===
-The physics of electromagnetic field is described mathematically by the Maxwell's equations<ref>[http://maxwell.sze.hu/docs/C4.pdf M. Kuczmann - Potential Formulations in Magnetics Applying the Finite Element Method, Lecture Note, University of Győr, 2009.]</ref>. These equations can be written in differential as well as integral forms. For time-varying electromagnetic fields, the Maxwell's equations are as follows.
+The physics of electromagnetic field is described mathematically by the Maxwell's equations<ref>[http://maxwell.sze.hu/docs/C4.pdf M. Kuczmann - Potential Formulations in Magnetics Applying the Finite Element Method, Lecture Note, University of Győr, 2009.]</ref> <ref>[https://drive.google.com/file/d/1DmkzXA8BZzlSZTPkjWbI093_mion6ETw/view?usp=sharing M. Kuczmann - The Finite Element Method in Electromagnetics, Presentation, University of Győr, 2016.]</ref>. These equations can be written in differential as well as integral forms. For time-varying electromagnetic fields, the Maxwell's equations are as follows.
 ==== Differential Form ====

@@ Line 18: / Line 18: @@
 <blockquote>
 Electromagnetic theory forms the foundation of different physical fields and phenomenon. It is used to explain many wave phenomena like propagation, reflection, refraction, diffraction and scattering. In the following, we shall review the fundamental aspects of the electromagnetic theory.
-</blockquote>
 === [https://en.wikipedia.org/wiki/Maxwell%27s_equations Maxwell's Equations] ===
 The physics of electromagnetic field is described mathematically by the Maxwell's equations<ref>[http://maxwell.sze.hu/docs/C4.pdf M. Kuczmann - Potential Formulations in Magnetics Applying the Finite Element Method, Lecture Note, University of Győr, 2009.]</ref>. These equations can be written in differential as well as integral forms. For time-varying electromagnetic fields, the Maxwell's equations are as follows.
 ==== Differential Form ====
 [[File:James Clerk Maxwell.png|250px|thumb|alt=James Clerk Maxwell (1831–1879).|James Clerk Maxwell (1831–1879).]]
@@ Line 100: / Line 100: @@
 The Maxwell's equations are general and hold for fields with arbitrary time dependence in any medium and at any location.
 === Constitutive Relations ===
 Equations which describe the relationship between field quatities called ''constitutive relations''. Constitutive relations are generally nonlinear, that is the permeability <math>\mu</math>, the conductivity <math>\sigma</math> and the permebility <math>\varepsilon</math> depend on the appropriate field quatities,
@@ Line 197: / Line 195: @@
 == Electromagnetics Models - Static Fields ==
 In the simplest case the time variation of the field quantities can be neglected, i.e. <math>\partial/\partial t = 0</math>. Electrostatic fields are usually produced by static electric charges, whereas static magnetic fields are due to the motion of electric charges with uniform velocity (direct current).
 === Static Magnetic Field ===
 The basic laws of static magnetic fields are Maxwell-Ampére law
@@ Line 219: / Line 217: @@
 ::<math>\nabla\times\left(\frac{1}{\mu}\nabla\times\vec{A}\right)=\vec{J}</math>.
 === Electrostatic Field ===
 The two fundamental laws governing these electrostatic fields are Gauss's law,