Pre-Quantum Electrodynamics

• PM 3.3

Suppose that we have two solutions to the Poisson equation, \(\phi_1 ({\bf r})\) and \(\phi_2 ({\bf r})\), within a certain volume \({\cal V}\) (which can contain charges).

Defining \(\Phi = \phi_1 - \phi_2\), we see that \(\Phi\) manifestly obeys Laplace within \({\cal V}\), \({\boldsymbol \nabla}^2 \Phi = 0\).

We can now use Green's first identity Green1 to shed some light on the boundary problem for the electrostatic potential. Namely, let us put \(\phi = \psi = \Phi\). This yields \[ \int_{\cal V} d\tau \left( \Phi {\boldsymbol \nabla}^2 \Phi + {\boldsymbol \nabla} \Phi \cdot {\boldsymbol \nabla} \Phi \right) = \oint_{\cal S} da ~\Phi \frac{\partial \Phi}{\partial n}. \] The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace. The right-hand side can be made to vanish if \(\Phi\) obeys either

\begin{equation} \left.\Phi\right|_{\cal S} = 0 \tag{Dirichlet}\label{Dirichlet} \end{equation}

or

\begin{equation} \left.\frac{\partial \Phi}{\partial n}\right|_{\cal S} = 0 \tag{Newmann}\label{Newmann} \end{equation}

boundary conditions on each individual boundary surface. In those cases, we are left with \[ \int_{\cal V} d\tau \left|{\boldsymbol \nabla} \Phi \right|^2 = 0, \] and since the integrand is strictly greater than or equal to zero everywhere, the only possibility is that it is zero everywhere, so we get the local condition \[ \longrightarrow {\boldsymbol \nabla} \Phi = 0. \] \(\Phi\) is thus constant. For Dirichlet boundary conditions, \(\Phi = 0\) throughout \({\cal V}\), and thus \(\phi_2 = \phi_1\), i.e. the solution is unique. For Neumann boundary conditions, the solution is unique apart from an unimportant additive constant.

We can thus finally state the

Note that these types of boundary conditions can be mixed, i.e. Dirichlet on some surfaces, Neumann on others).

Existence of solutions: this is another matter. Intuitively, from our first case: the solution always exists for Dirichlet boundary conditions. For Neumann, some self-consistency restrictions apply.

Link to earlier cases: the previous "known boundary potential" case is thus the case of Dirichlet boundary conditions. The "known boundary charge" is the case of Neumann boundary conditions.

Note on presentations of uniqueness theorem(s) in various books: we have used Green's identity to provide a general statement on uniqueness. Reading other books, you might be misled into thinking that there are numerous cases and corollaries and that things are messy. Rest assured, because we can state the uniqueness theorem on uniqueness theorems: