Electromagnetism lecture uniqueness theorem youtube. Uniqueness theorems in electrostatics laplace and poisson. This video lecture covers the in class proceedings in electromagnetism taught to the final third year class physics. Under what conditions, there exists a unique solution to 1.
The solution to the laplace equation in some volume is uniquely determined if the potential voltage is specified on the boundary surface. What links here related changes upload file special pages permanent. The theorem allows us to make predictions on the length of the interval that is h is less than or equal to the smaller of the numbers a and bm. The existence and uniqueness theorem of the solution a. The existence and uniqueness theorem of the solution a first order linear equation initial value problem does an initial value problem always a solution.
Two methods for solving electrostatic problems with azimuthal. Electrostatictheorems university of texas at austin. In these notes i prove several important theorems concerning the electrostatic potential v x,y,z, namely the earnshaw theorem, the meanvalue theorem, and two uniqueness theorems. Recall that our previous proof of this was rather involved, and was also not particularly rigorous see sect. 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. The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. Suppose we have two solutions of laplaces equation, vr v r12 and g g, each satisfying the same boundary conditions, i. The answer is yes, there is only one solution, and it is unique. The potential v in the region of interest is governed by the poisson equation.
Existence and uniqueness theorem jeremy orlo theorem existence and uniqueness suppose ft. If a solution of laplaces equation can be found that satisfies the. The uniqueness theorem university of texas at austin. We know that the interior surface of the conductor is at some constant potential. One immediate use of the uniqueness theorem is to prove that the electric field inside an empty cavity in a conductor is zero. Here we concentrate on the solution of the rst order ivp y0 f x. Pdf existence and uniqueness theorem for set integral. The uniqueness theorem actually stems from differential equation mathematics. This means also that if you found a solution that fulfils these conditions, it is the only solution you have. The solution of the poisson equation inside v is unique if either dirichlet or neumann boundary condition on s is satisfied. There is no reason to introduce scalar potential only.
In the case of electrostatics, this means that there is a unique electric field derived from a potential function satisfying poissons equation under the boundary conditions. The following theorem states a precise condition under which exactly one solution would always exist for a given initial value problem. Uniqueness theorem for poissons equation wikipedia. Proof we suppose that two solutions and satisfy the same boundary conditions. Differential equations existence and uniqueness theorem. Pdf uniqueness theorem, theorem of reciprocity, and. Pdf the method of image charges relies on the proven uniqueness of. In fact, although 1 is consistent with the existing uniqueness theorem in electrostatics that requires normal.