Chapter3 Boundary-ValueProblemsinElectrostaticsDifferential EguationsforElectricPotentialMethod of ImagesMethodofSeparationofVariables1.DifferentialEguationsforElectricPotential2. Method of lmage3. Method of Separation of Variables in RectangularCoordinates4.Methodof SeparationofVariablesinCylindricalCoordinates5.MethodofSeparationof VariablesinSphericalCoordinates
Chapter 3 Boundary-Value Problems in Electrostatics Differential Equations for Electric Potential Method of Images Method of Separation of Variables 1. Differential Equations for Electric Potential 2. Method of Image 3. Method of Separation of Variables in Rectangular Coordinates 4. Method of Separation of Variables in Cylindrical Coordinates 5. Method of Separation of Variables in Spherical Coordinates
1.DifferentialEquationsforElectricPotentialThe relationship between the electric potential o and the electricfieldintensityEisE=-VpTaking the divergence operation forboth sides of the aboveequationgivesV.E=-V?0In a linear, homogeneous, and isotropic medium, the divergenceofthe electricfieldintensityEisV.E-P8U7
1. Differential Equations for Electric Potential The relationship between the electric potential and the electric field intensity E is Taking the divergence operation for both sides of the above equation gives In a linear, homogeneous, and isotropicmedium, the divergence of the electric field intensity E is E = − 2 E = − E =
The differential equationforthe electricpotentialisV?p=-P8whichis called Poisson'sequationIn a source-freeregion,and the above equation becomesVβ=0whichis calledLaplace'sequationThe solution of Poisson's EquationIn infinite free space, the electric charge densityp(confined toin V produces the electric potentialgiven byp(r)4元8which is just the solution for Poisson's Equation in free spaceU7
The differential equation for the electric potential is = − 2 which is called Poisson’s equation. In a source-freeregion, and the above equation becomes 0 2 = which is called Laplace’s equation. The solution of Poisson’s Equation. V V − = d | | ( ) 4π 1 ( ) r r r r In infinite free space, the electric charge density confined to in V producesthe electric potential given by (r) which is just the solution for Poisson’sEquation in free space
Applying Green's function G(r, r') gives the general solution ofPoission'sequationo(n)=f,G(r,r)p(r2dv'+8f,[G(r, r)V'p(r)-p(r')v'G(r, r') dsForinfinite free space, the surface integralin the above equation willbecome zero,and Green's function becomes1Go(r, r') =-4元|r-r'|In the source-free region, the volume integral in the aboveequation will be zero. Therefore, the second surface integral isconsidered to be the solution of Poisson's equation in source-freeregion, or the integral solution of Laplace's equation in terms ofGreen'sfunction.UV
Applying Green’s function gives the general solutionof Poission’s equation G(r, r) r r r r r r S r r r r [ ( , ) ( ) ( ) ( , )] d d ( ) ( ) ( , ) − + = G G G V S V 4π | | 1 ( , ) 0 r r r r − G = For infinite free space, the surface integral in the above equation will become zero, and Green’s function becomes In the source-free region, the volume integral in the above equation will be zero. Therefore, the second surface integral is considered to be the solution of Poisson’s equation in source-free region, or the integral solution of Laplace’s equation in terms of Green’sfunction
An eguationin mathematicalphysicsis to describethe changes ofphysicalquantities with respectto space and time. For the specifiedregion and moment, the solution of an equation depends on the initialcondition and the boundary condition,respectively,and both are alsocalledthesolvingconditionUsuallythe boundary conditions are classifiedinto threetypes1.Dirichetboundary condition:The physical quantities on theboundariesarespecified2. Neumann boundary condition:The normalderivatives of thephysical quantities on the boundaries are given.3. Mixed boundary-value condition:The physical quantities onsome boundaries are given, and the normal derivatives of the physicalquantities are specified on the remaining boundaries
An equation in mathematical physics is to describe the changes of physical quantities with respect to space and time. For the specified region and moment, the solution of an equation depends on the initial conditionand the boundary condition, respectively, and both are also called the solving condition. 2. Neumann boundary condition: The normal derivatives of the physical quantities on the boundaries are given. 3. Mixed boundary-value condition: The physical quantities on some boundaries are given, and the normal derivatives of the physical quantities are specified on the remaining boundaries. 1. Dirichetboundary condition: The physical quantities on the boundaries are specified. Usually the boundary conditions are classified into three types: