In the case of electrostatics, this means that there is a unique electric field. We assert that the two solutions can at most differ by a constant. Suppose that, in a given finite volume bounded by the closed surface, we have. Suppose that the value of the electrostatic potential is specified at every point on the surface of this volume. The surface of the photoconductor is initially charged uniformly as in figure 6. The solution of the poisson equation inside v is unique if either dirichlet or neumann boundary condition on s is satisfied. Uniqueness theorem for poissons equation wikipedia. Suppose we have two solutions of laplaces equation, vr v r12 and g g, each satisfying the same boundary conditions, i. Introduction in these notes, i shall address the uniqueness of the solution to the poisson equation. As we have seen in previous lectures, very often the primary task in an electrostatics problem.
The solution to the laplace equation in some volume is uniquely determined if the potential voltage is specified on the boundary surface. Existence and uniqueness theorem jeremy orlo theorem existence and uniqueness suppose ft. Pdf existence and uniqueness theorem for set integral. If for some r 0 a power series x1 n0 anz nzo converges to fz for all jz zoj potential boundary value problems 2. The first uniqueness theorem implies that simion can calculate unique values of nonelectrode point potentials within any volume that is. Proof on a uniqueness theorem in electrostatics physics. Boundary conditions for electric field part 2 proof electrostatics for b. First, suppose that some volume v is surrounded by a conducting surface s, for instance, a metal foil, and sources of the field e 0 are located outside this volume fig. Uniqueness theorems in electrostatics laplace and poisson. First uniqueness theorem simion 2019 supplemental documentation.
The uniqueness theorem for poissons equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. This line was enough for me get a feel of uniqueness theorem, understand its importance and. Aug 10, 2019 the solution to laplaces equation in some volume is uniquely determined if the equation is specified on the boundary. Existence and uniqueness theorem for setvalued volterra.
Uniqueness theorem there is a uniqueness theorem for laplaces equation such that if a solution is found, by whatever means, it is the solution. Then we can choose a smaller rectangle ras shown so that the ivp dy dt ft. If you know one way, you can be sure that nature knows no other way this was what our physics teacher told us when he was teaching uniqueness theorem. Chapter 2 poissons equation university of cambridge.
The solution to laplaces equation in some volume is uniquely determined if the equation is specified on the boundary. Existence and uniqueness theorem in a situation where the yderivative is unbounded hot network questions are matthew 11. The second uniqueness theorem states that the electric field is uniquely determined if the total charge on each conductor is given and the charge distribution in the regions between the conductors is known. Uniqueness theorem definition of uniqueness theorem by. The first uniqueness theorem states that in this case the solution of laplaces equation is uniquely defined. We include appendices on the mean value theorem, the. A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column, that is, if and only if an echelon form of the augmented matrix has no row of the form 0 0b, with b 6d0. Uniqueness of solutions to the laplace and poisson equations 1. As we know, due to electrostatic induction, positive and negative charges arise on the external surface of the. The uniqueness theorem university of texas at austin.
More precisely, the solution to that problem has a discontinuity at 0. For laplaces equation, if we have the boundaries of a region specified, we have exactly one solution inside the. School of mathematics, institute for research in fundamental sciences ipm p. That is, suppose that there is a region of space of volume v and the boundary of that surface is denoted by s. More applications of vector calculus to electrostatics. These theorems are also applicable to a certain higher order ode since a higher order ode can be reduced to a system of rst order ode. Poissons equation the uniqueness theorem we have already seen the great value of the uniqueness theorem for poissons equation or laplaces equation in our discussion of helmholtzs theorem see sect. This result leads to the following uniqueness theorem which can be improved making weaker some hypotheses on the behaviour of the function on the regular boundary. The existence and uniqueness theorem of the solution a. Then, where n is the outwardly directed unit normal to the surface at that point, da is an element of surface area, and is the angle between n and e, and d is the element of solid angle.
Alexandrovs uniqueness theorem of threedimensional polyhedra. The curl of an electrostatic curl f da for any surface a 0 curl in cartesian coordinates 1. Uniqueness theorems bibliography using the helmholtz theorem and that b is divergenceless, the magnetic eld can be expressed in terms of a vector potential, a. Such a uniqueness theorem is useful for two reasons. The first uniqueness theorem can only be applied in those regions that are free of charge and surrounded by a boundary with a known potential not necessarily. Uniqueness theorem there are several methods of solving a given problem analytical, graphical, numerical, experimental, etc.
The proof of the second uniqueness theorem is similar to the proof of the first uniqueness theorem. But the authors have aimed the book at an audience which is not expected to have studied uniform convergence as described in the preliminary. In mathematics, a uniqueness theorem is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. For any radius 0 theorem 2 existence and uniqueness theorem 1.
Pdf uniqueness theorem, theorem of reciprocity, and. The existence and uniqueness theorem are also valid for certain system of rst order equations. The existence and uniqueness theorem of the solution a first. To proof the first uniqueness theorem we will consider what happens when there are two solutions v 1 and v 2 of laplaces equation in the volume shown in figure 3. More details can be found in griffiths book introduction to electrodynamics. The uniqueness theorem sheds light on the phenomenon of electrostatic induction and the shielding effect. Uniqueness theorem, theo rem of reciprocity, and ei genv alue pro blems 483 3 general theorems let us consider a b o dy with the volume v b ounded by a sur face s at tim e t 0.
You can make the solution unique if you specify further boundary conditions, but the theorem is more technical. Uniqueness theorems consider a volume see figure 3. Given some boundary conditions, do we have enough to find exactly 1 solution. The potential v in the region of interest is governed by the poisson equation. But the authors have aimed the book at an audience which is not expected to have studied uniform convergence as. If a linear system is consistent, then the solution set contains either. Uniqueness theorem an overview sciencedirect topics. Laplaces equation and poissons equation in this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for poissons equation. At undergraduate level, it is interesting to work with the moment generating function and state the above theorem without proving it. A uniqueness theorem or its proof is, at least within the mathematics of differential equations, often combined with an existence theorem or its proof to a combined existence and uniqueness theorem e. What is an intuitive explanation of the second uniqueness. Proof on a uniqueness theorem in electrostatics physics forums. The first uniqueness theorem is the most typical uniqueness theorem for the laplace equation.
Pdf existence and uniqueness theorem for set integral equations. Sep 12, 2012 given some boundary conditions, do we have enough to find exactly 1 solution. If for some r 0 a power series x1 n0 anz nzo converges to fz for all jz zoj theorem on integration of power series. We shall show in this section that a potential distribution obeying poissons equation is completely specified within a volume v if the potential is specified over the surfaces bounding that volume. Study of electricity in which electric charges are static i. The electric field at a point on the surface is, where r is the distance from the charge to the point.
1446 657 634 239 664 1097 1078 159 1574 1136 1327 766 678 1531 1211 283 14 1525 711 74 589 1211 525 1071 787 1404 200 1079 856 1465 1272