[대학원-전자기학]문제풀이

Problem Prove the symmetry property of the Green’s function satisfying the Dirichlet boundary condition $$G(\vec{x},\vec{x}^{\prime}) = G(\vec{x}^{\prime}, \vec{x})$$

Refer to Jackson Problem (1.14)

Answer

Green function $G(\vec{x},\vec{x}^{\prime})$ satisfies Dirichlet boundary conditions; bound region $\Omega$ with boundary $d\Omega$, $G(\vec{x},\vec{x}^{\prime})=0 \quad \forall x^{\prime} \in \partial \Omega$

$$\int_V (\phi \nabla^2 \psi - \psi \nabla^2 \phi)d^3x = \oint_s \left[ \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi }{\partial n}\right]da$$

Substitute $\phi = G(\vec{x},\vec{y})$ and $\psi = G(\vec{x}^{\prime},\vec{y})$.

$$\begin{aligned} \int_{\Omega} \left[ G(\vec{x},\vec{y})\nabla^2 G(\vec{x}^{\prime},\vec{y}) - G(\vec{x}^{\prime},\vec{y}) \nabla^2 G(\vec{x},\vec{y}) \right]d^3y =\nonumber \\ \int_{\partial \Omega}\left[ G(\vec{x}, \vec{y})\frac{\partial }{\partial n}G(\vec{x}^{\prime},\vec{y}) - G(\vec{x}^{\prime},\vec{y}) \frac{\partial}{\partial n} G(\vec{x},\vec{y}) \right]da \nonumber\end{aligned}$$

$G(\vec{x}, \vec{x}^{\prime})$ satisfies Dirichlet B.C. Hence $RHS=0$; $\Phi$ is known on the surface and $F$ can be chosen to make $G_D(\vec{x},\vec{x}^{\prime}=0)$ Since $$\nabla^2 G(\vec{x},\vec{y}) = -4\pi \delta^3(\vec{x}-\vec{y})$$

$$\begin{aligned} \int_{\Omega} \left[ G(\vec{x},\vec{y})\nabla^2 G(\vec{x}^{\prime},\vec{y}) - G(\vec{x}^{\prime},\vec{y}) \nabla^2 G(\vec{x},\vec{y}) \right]d^3y = \nonumber \\ \int_{\Omega}\left[ -4\pi G(\vec{x},\vec{y}) \delta^3(\vec{x}^{\prime} - \vec{y})+G(x^{\prime},\vec{y}) 4\pi \delta^3(\vec{x}-\vec{y}) \right]d^3y \nonumber \\\end{aligned}$$

Finally

$$G(\vec{x},\vec{x}^{\prime}) = G(\vec{x}^{\prime}, \vec{x})$$

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