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