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:
For any mathematical physics equation, the existence, the stabilityand the uniqueness ofthe solutions need to be investigated.The existence of the solution is that whether the equation has asolution or not for the given condition of the solutionThe stability of the solution refers to whether the solution ischanged substantially when the condition or the solution is changedslightly.The uniqueness of the solution is whether the solution is unique ornotforthe prescribed condition ofthe solutionElectrostatic fields exist in nature, and the existence of the solutionof the differential equations for the electric potentialis undoubtedThe conditions of the solution are derived from measurements, theyare subject to inaccuracy. Therefore, the stability of the solution haspracticalsignificance
For any mathematical physics equation, the existence, the stability, and the uniqueness of the solutions need to be investigated. The conditions of the solution are derived from measurements, they are subject to inaccuracy. Therefore, the stability of the solution has practicalsignificance. The uniqueness of the solution is whether the solution is unique or not for the prescribed condition of the solution. The stability of the solution refers to whether the solution is changed substantially when the condition or the solution is changed slightly. The existence of the solution is that whether the equation has a solution or not for the given condition of the solution. Electrostatic fields exist in nature, and the existence of the solution of the differential equationsfor the electric potential is undoubted
The stability of Poisson's and Laplace's equations have been provedin mathematics, and the uniqueness ofthe solution of the differentialequationsforthe electricpotentialcanbeprovedalso.In many practicalsituations, the boundary for the electrostatic fieldis on a conducting surface.In such cases,the electric potentialon theboundary is given by the first type of boundary condition, and theelectric charge is given by the second type of boundary condition.Therefore, the solution for the electrostatic fieldis unigue when thecharge is specified on the surface of the conducting boundary.For electrostatic fields with conductors as boundaries, the fieldmay be given uniquely when the electric potential, its normalderivative, or the charges is given on the conducting boundaries.Thatis the uniqueness theorem for solutions to problems on electrostaticfields
In many practical situations, the boundary for the electrostatic field is on a conducting surface. In such cases, the electric potential on the boundary is given by the first type of boundary condition, and the electric charge is given by the second type of boundary condition. Therefore, the solution for the electrostatic field is unique when the charge is specified on the surface of the conducting boundary. For electrostatic fields with conductors as boundaries, the field may be given uniquely when the electric potential , its normal derivative, or the charges is given on the conducting boundaries. That is the uniqueness theorem for solutions to problems on electrostatic fields. The stability of Poisson’s and Laplace’s equations have been proved in mathematics, and the uniqueness of the solution of the differential equationsfor the electric potential can be proved also
2.MethodoflmageEssence:The effect of the boundaryis replaced by one orseveralequivalentcharges,and the originalinhomogeneousregionwith a boundary becomes an infinitehomogeneous space.Basis : The principle of uniqueness. Therefore, these chargesshould not change the originalboundary conditions.Theseequivalent charges are at the image positions of the original chargesand are calledimage charges,and this method is called the methodofimages.Key : To determine the values and the positions of the imagecharges.Restriction:Theseimage charges maybe determined onlyforsome special boundaries and charges with certain distributionsuV
2. Method of Image Essence: The effect of the boundary is replaced by one or several equivalent charges, and the original inhomogeneous region with a boundary becomes an infinite homogeneousspace. Basis：The principle of uniqueness. Therefore, these charges should not change the original boundary conditions. These equivalent charges are at the image positions of the original charges, and are called image charges, and this method is called the method of images. Key：To determine the values and the positions of the image charges. Restriction：These image charges may be determined only for some special boundaries and charges with certain distributions
(1)Apoint electric charge and an infinite conductingplaneDLDielectricDielectricSDielectricConductorThe effect of the boundary is replaced by a point charge at theimage position, while the entire space becomes homogeneous withpermittivity s, then the source of electric potential at any point Pwill be due to the charges q and q',qqD:4元r4元rConsidering the electric potentialofan infinite conductingplaneis zero, we have q'= -q.UV
（1）A point electric charge and an infinite conducting plane Dielectric Conductor q r P The effect of the boundary is replaced by a point charge at the image position, while the entire space becomes homogeneous with permittivity , then the source of electric potential at any point P will be due to the charges q and q', r q r q = + 4π 4π Considering the electric potential of an infinite conducting plane is zero, we have . q = −q Dielectric q r P h h r q Dielectric
The distribution of the electric field lines and the equipotentialsurfaces are the same as that of an electric dipole in the upper halfspace.The electric field lines are perpendicularto the conductingsurfaceeverywhere, which has zero potential
The distribution of the electric field lines and the equipotential surfaces are the same as that of an electric dipole in the upper halfspace. The electric field lines are perpendicular to the conducting surface everywhere, which has zero potential. z