1、Maxwells equations in static electric field,Boundary condition in static electric field,Ohms law,Maxwells equations for Steady electric currents,Maxwells equations in Static Magnetic Fields,Boundary-value problems,Boundary-value problems is used to solve Possions equation or Laplace equation.,Types

2、of Boundary-value problems,the first (Dirichlet problems ):,the third (mixed ):,the second (Neumann problems ):,natural Boundary-values,periodical Boundary values,Interface of two medias,（the first boundary-value problems ）,（the first boundary-value problems ）,Uniqueness Theorem in static field,Meth

3、ods that can lead to same Boundary-value is equivalent.,If or is specified on surface , the solution of Possions equation or Laplaces equation is unique.,proof by reduction to absurdity :,Inside V :,Or,or,to proof :,let,On S : (set Boundary-value conditions ),For first Green identity:,For the first

4、：,For the third ：,For the second ：,Image Method,Non-Uniform inductive charges,Equivalent charge,For complex integral area, Gausss law is hard to solve the electric problems,Example,q,q,Image method : use simple distributed equivalent charges to substitute the non-uniform distributed charges.,key pro

5、blem：how to make sure the solution is correct ?,Example,Equivalent charge,Non-Uniform inductive charges,Based on the Uniqueness Theorem :,If boundary-value is specified on surface, the solution is unique. so keep boundary-value invariant, the solution is invariant too.,or,or,Two rules to find equiva

6、lents charges,1 equivalents charges is distributed outside of integral area,2 Change the charge number or Charge position or charge quantity to keep the same boundary-value of integral area.,/,Example: one discrete charge,boundary-value:,boundary-value:,Image charge:,The solution area: z0,The total

7、inductive charges :,Example: linear distributed charges,Example,Example: Conducting sphere,to find ：,Surface of Sphere,Total inductive charges on surface of the sphere:,Method of separation of variables,for k2,For 2D :,for -k2,Example,For (0 , y)0、 (a , y)0，so,In engineering, sinusoidal time functio

8、ns are easy to generate; arbitrary periodic time functions can be expanded into Fourier series of harmonic sinusoidal components; and transient non-periodic functions can be expressed as Fourier integrals.,Time-Harmonic Fields,Maxwells Equations in Frequency Domain,Coulombs Law,Amperes Law,Faradays

9、Law,Uniqueness Theorem,Uniqueness Theorem: The solution is unique in the lossy medium if the tangential electric field, or tangential magnetic field, or the tangential electric field in the part of surface and the tangential magnetic field in the other part of surface are Determined. There are many resonant solutions in lossless medium even if the boundary condition is specified.,Equivalence Principles,Reciprocity Theorem,Suppose a set of source Ja and Ma, which generates a field Ea and Ha another set of source Jb