Question

Find the Green's function \(G\left(\mathbf{r}, \mathbf{r}_{0}\right)\) in the half-space \(z>0\) for the solution of \(\nabla^{2} \Phi=0\) with \(\Phi\) specified in cylindrical polar coordinates \((\rho, \phi, z)\) on the plane \(z=0\) by $$ \Phi(\rho, \phi, z)= \begin{cases}1 & \text { for } \rho \leq 1 \\ 1 / \rho & \text { for } \rho>1\end{cases} $$ Determine the variation of \(\Phi(0,0, z)\) along the \(z\)-axis.

Short answer

\(\Phi(0,0,z) = 1\)

Step-by-step solution

Understand the Problem Setup

The goal is to find the Green's function, denoted as \(G(\mathbf{r}, \mathbf{r}_{0})\), which satisfies the Laplace equation, \(abla^{2} \Phi=0\), in the half-space \(z>0\). The boundary condition for \(\Phi\) is provided on the plane \(z=0\) in cylindrical coordinates \((\rho, \phi, z)\).

Define the Boundary Conditions

The boundary condition at \(z=0\) is given by:\[ \Phi(\rho, \phi, 0)= \begin{cases} 1 & \text{for } \rho \leq 1 \ \frac{1}{\rho} & \text{for } \rho > 1 \end{cases} \]

Set Up the Green’s Function

Green’s function for the Laplacian in free space is:\[ G_{0}(\mathbf{r},\mathbf{r}_{0}) = -\frac{1}{4\pi|\mathbf{r}-\mathbf{r}_{0}|} \]For the half-space \(z>0\), we must account for the boundary at \(z=0\), modifying this to:\[ G(\mathbf{r}, \mathbf{r}_{0}) = -\frac{1}{4\pi|\mathbf{r}-\mathbf{r}_{0}|} + \frac{1}{4\pi|\mathbf{r}-\mathbf{r}_{0}'|} \] where \(\mathbf{r}_{0}'\) is the image point of \(\mathbf{r}_{0}\) about the plane \(z=0\).

Apply Boundary Conditions

For \(\mathbf{r} = (\rho, \phi, 0)\) and \(\Phi(\rho, \phi, 0)\) is known, we ensure:\[ \Phi(\rho, \phi, 0) = \int_{\mathbf{r}_{0}} G(\mathbf{r},\mathbf{r}_{0}) \delta(\mathbf{r}_{0}) \,d^3\mathbf{r}_{0} \]

Determine \(\Phi(0,0,z)\)

Since \(\Phi\) on the boundary \(z=0\) is rotationally symmetric about the origin, extend this symmetry in \(z>0\). Consequently, for \(\rho=0\):\[ \Phi(0, 0, z) = \Phi(\rho, \phi, z) \text{ for } \rho = 0 \text{ and any } z \]Evaluating along the \(z\)-axis, use the boundary conditions:For \(\rho \leq 1\), at \(\rho=0\), initially \(abla^2 \to 0\) leading to maintaining \(abla^2 G\).

Solve for \(\Phi(0,0,z)\)

Considering the boundary effect and symmetrical placement, \(abla^2G=0\) all points:\[ \Phi(0,0,z) = 1 \text{ along } z \] as influence boundary.

Key concepts

Laplace Equation

The Laplace equation, represented as \(abla^2 \, \theta = 0\), is a second-order partial differential equation. It describes the behavior of scalar fields like gravitational potentials, electric potentials, and fluid velocity potentials in different areas of physics and engineering. Solving the Laplace equation helps us to find steady-state solutions without any time dependence. In simple terms, it models processes where the field does not change over time. This often involves calculating the values inside a region based on values specified at the boundaries. For instance, if we know the temperature on the walls of a container, we can use the Laplace equation to find the temperature distribution inside the container. In the given problem, we aim to find a potential function \(abla^2 \, \theta = 0\) by utilizing boundary conditions.

Boundary Conditions

Boundary conditions are constraints necessary for solving differential equations, like the Laplace equation. They define the behavior of a solution on the boundaries of the domain. In the context of the problem, the boundary conditions are given on the plane \(z=0\) and are specified in cylindrical polar coordinates \((\rho, \phi, z)\). They state that the potential \(\theta(\rho, \phi, z)\) is 1 for \(\rho \leq 1\) and \(1/\rho\) for \(\rho > 1\). These conditions help us understand how the function behaves on the boundary so we can accurately determine its behavior in the entire region. Boundary conditions can be categorized as Dirichlet (specifying the function value), Neumann (specifying derivative values), or mixed (a combination of both). Here, we use Dirichlet boundary conditions to set explicit values of the potential function on the boundary.

Cylindrical Coordinates

Cylindrical coordinates \((\rho, \phi, z)\) are a three-dimensional coordinate system. They are useful when dealing with problems exhibiting symmetry around an axis, such as the cylindrical symmetry in this problem. Here, \(\rho\) is the radial distance from the axis, \(\phi\) is the azimuthal angle, and \(z\) is the height. This system simplifies the Laplace equation and boundary conditions, especially when dealing with problems involving cylinders or circular regions. For example, the boundary conditions are naturally given in terms of \(\rho\). The cylindrical coordinates modify the Laplace equation to \(\frac{1}{\rho} \frac{\partial}{\partial \rho}(\rho \frac{\partial \theta}{\partial \rho}) + \frac{1}{\rho^2} \frac{\partial^2 \theta}{\partial \phi^2} + \frac{\partial^2 \theta}{\partial z^2} = 0\). This adapted formulation applies directly to the shape and symmetry of the physical setup.

Half-Space Solutions

Half-space solutions involve solving problems constrained to one half of the space, such as \(z > 0\). They are common in physics and engineering for modeling phenomena like heat flow or electrostatics above a conducting plane. The concept uses image points to help satisfy boundary conditions. In our problem, we modify the free-space Green's function by introducing an image point to account for the boundary at \(z = 0\). The Green's function becomes \(G(\mathbf{r}, \mathbf{r}_0) = -\frac{1}{4\pi|\mathbf{r}-\mathbf{r}_0|} + \frac{1}{4\pi|\mathbf{r}-\mathbf{r}_0'|}\), where \(\mathbf{r}_0'\) is the reflection of \(\mathbf{r}_0\) across the \(z = 0\) plane. This modification ensures that the boundary conditions are satisfied, allowing us to find the potential \(\theta(0,0,z)\) along the \(z\)-axis. Utilizing these methods ensures that solutions are consistent and physically meaningful over the specified domain.