静电场的微分方程与解的唯一性(双语)

Poission’s equation
(r)
V G(r,
r) ( r) dV
S [G(r, r)(r) (r)G(r, r)] dS
For infinite free space, the surface integral in the above equation will
become zero, and Green’s function becomes
equation gives
E 2
In a linear, homogeneous, and isotropic medium, the divergence of the electric field intensity E is
E
The differential equation for the electric potential is
2
which is called Poisson’s equation.
In a source-free region, and the above equation becomes
2 0
which is called Laplace’s equation.
The solution of Poisson’s InEinqfuinatitioenfr. ee space, the electric charge density
Usually the boundary conditions are classified into three types:
1. Dirichet boundary condition: The physical quantities on the boundaries are specified.
Cha来自百度文库ter 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
For any mathematical physics equation, the existence, the stability, and the uniqueness of the solutions need to be investigated.
in V produces the electric potential given by
(r)
1 4π
V
|
(r ) r r
|dV
(rco)nfined to
which is just the solution for Poisson’s Equation in free space.
Applying Green’s function G(r, r) gives the general solution of
1. Differential Equations for Electric Potential
The relationship between the electric potential and the electric field intensity E is
E
Taking the divergence operation for both sides of the above
G0 (r,
r)
1 4π | r r |
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’s function.
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 condition and 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.