4.1 Poisson's equation

Poisson's equation is a key concept in potential theory, describing how potential fields behave with sources or sinks. It connects the Laplacian of a potential function to a source term, enabling us to solve for electric fields, gravitational fields, and more.

Understanding Poisson's equation is crucial for tackling real-world problems in physics and engineering. By mastering its derivation, boundary conditions, and solution methods, we gain powerful tools for analyzing complex systems and predicting their behavior.

Definition of Poisson's equation

• Poisson's equation is a partial differential equation (PDE) that describes the behavior of a potential function in the presence of a source term
• It relates the Laplacian of the potential function to the source term, which represents the density of the source or sink of the potential field
• The solution to Poisson's equation gives the potential function, which can be used to calculate various physical quantities such as electric fields, gravitational fields, or fluid velocities

Laplace operator

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• The Laplace operator, denoted as $\nabla^2$, is a second-order differential operator that measures the divergence of the gradient of a function
• In Cartesian coordinates, the Laplace operator is defined as $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$
• The Laplace operator appears on the left-hand side of Poisson's equation, acting on the potential function

Source term

• The source term, usually denoted as $f(x, y, z)$ or $\rho(x, y, z)$, represents the density of the source or sink of the potential field
• In electrostatics, the source term is the charge density (volume charge density)
• In gravitation, the source term is the mass density multiplied by the gravitational constant

Solution as potential function

• The solution to Poisson's equation is the potential function, often denoted as $\phi(x, y, z)$ or $V(x, y, z)$
• The potential function describes the potential energy per unit charge (electrostatics) or mass (gravitation) at each point in space
• Once the potential function is known, various physical quantities can be derived from it, such as electric fields ($\vec{E} = -\nabla \phi$) or gravitational fields ($\vec{g} = -\nabla V$)

Derivation of Poisson's equation

• Poisson's equation can be derived from fundamental physical laws, such as Gauss's law in electrostatics or Newton's law of universal gravitation
• The derivation involves applying the divergence theorem to the flux of the potential field through a closed surface and relating it to the enclosed source or sink

From Gauss's law

• In electrostatics, Gauss's law states that the flux of the electric field through any closed surface is proportional to the total electric charge enclosed within that surface
• Mathematically, Gauss's law is expressed as $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$, where $\vec{E}$ is the electric field, $d\vec{A}$ is the area element, $Q_{enc}$ is the enclosed charge, and $\epsilon_0$ is the permittivity of free space
• Applying the divergence theorem to Gauss's law and using the relation between the electric field and the potential $(\vec{E} = -\nabla \phi)$ leads to Poisson's equation: $\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$, where $\rho$ is the charge density

In electrostatics

• In electrostatics, Poisson's equation relates the electric potential $\phi$ to the charge density $\rho$
• The equation takes the form $\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$, where $\epsilon_0$ is the permittivity of free space
• Solving Poisson's equation in electrostatics allows us to determine the electric potential and electric field distribution for a given charge configuration

In gravitation

• In Newtonian gravitation, Poisson's equation relates the gravitational potential $V$ to the mass density $\rho_m$
• The equation takes the form $\nabla^2 V = 4\pi G \rho_m$, where $G$ is the gravitational constant
• Solving Poisson's equation in gravitation enables us to calculate the gravitational potential and gravitational field for a given mass distribution

Boundary conditions

• Boundary conditions specify the values or behavior of the potential function at the boundaries of the domain in which Poisson's equation is being solved
• They are essential for obtaining a unique solution to Poisson's equation and ensuring that the solution is physically meaningful
• The three main types of boundary conditions are Dirichlet, Neumann, and mixed boundary conditions

Dirichlet boundary conditions

• Dirichlet boundary conditions, also known as fixed boundary conditions, specify the values of the potential function on the boundary of the domain
• Mathematically, Dirichlet boundary conditions are expressed as $\phi(x, y, z) = f(x, y, z)$ on the boundary, where $f(x, y, z)$ is a known function
• Examples of Dirichlet boundary conditions include specifying the electric potential on the surface of a conductor or the temperature on the walls of a heat-conducting object

Neumann boundary conditions

• Neumann boundary conditions, also called flux boundary conditions, specify the normal derivative of the potential function on the boundary of the domain
• Mathematically, Neumann boundary conditions are expressed as $\frac{\partial \phi}{\partial n} = g(x, y, z)$ on the boundary, where $\frac{\partial \phi}{\partial n}$ denotes the normal derivative and $g(x, y, z)$ is a known function
• Examples of Neumann boundary conditions include specifying the electric field on the surface of a conductor or the heat flux on the walls of a heat-conducting object

Mixed boundary conditions

• Mixed boundary conditions, also known as Robin boundary conditions, involve a combination of Dirichlet and Neumann boundary conditions
• Mathematically, mixed boundary conditions are expressed as $a\phi + b\frac{\partial \phi}{\partial n} = c$ on the boundary, where $a$, $b$, and $c$ are known constants or functions
• Mixed boundary conditions are useful when modeling situations where the potential function and its normal derivative are related on the boundary (convective heat transfer or leaky dielectrics)

Green's function approach

• The Green's function approach is a powerful method for solving Poisson's equation by expressing the solution as an integral involving the source term and a special function called the Green's function
• The Green's function is a fundamental solution to the corresponding homogeneous equation (Laplace's equation) with a point source, satisfying the appropriate boundary conditions

Definition of Green's function

• The Green's function, denoted as $G(x, y, z; x', y', z')$, is a function that satisfies the following properties:
  1. It is a solution to the homogeneous equation $\nabla^2 G = 0$ everywhere except at the source point $(x', y', z')$
  2. It satisfies the boundary conditions of the problem
  3. It has a singularity at the source point, typically of the form $\frac{1}{r}$ in 3D, where $r$ is the distance between $(x, y, z)$ and $(x', y', z')$
• The Green's function depends on the geometry of the domain and the type of boundary conditions

Derivation of Green's function

• The derivation of the Green's function involves solving the homogeneous equation $\nabla^2 G = 0$ with a point source, typically using the method of separation of variables or Fourier transforms
• The solution must satisfy the boundary conditions of the problem and have the appropriate singularity at the source point
• The derivation may involve the use of special functions, such as Bessel functions or Legendre polynomials, depending on the geometry of the domain

Green's function in different dimensions

• The form of the Green's function depends on the dimensionality of the problem:
  • In 1D, the Green's function is typically a piecewise linear function with a jump discontinuity at the source point
  • In 2D, the Green's function is often a logarithmic function, such as $G(x, y; x', y') = -\frac{1}{2\pi} \ln(r)$, where $r$ is the distance between $(x, y)$ and $(x', y')$
  • In 3D, the Green's function is usually a radial function, such as $G(x, y, z; x', y', z') = \frac{1}{4\pi r}$, where $r$ is the distance between $(x, y, z)$ and $(x', y', z')$
• Once the Green's function is known, the solution to Poisson's equation can be expressed as an integral: $\phi(x, y, z) = \int_\Omega G(x, y, z; x', y', z') f(x', y', z') dx' dy' dz'$, where $\Omega$ is the domain and $f(x', y', z')$ is the source term

Method of images

• The method of images is a technique for solving Poisson's equation in the presence of boundaries by replacing the boundaries with fictitious sources or sinks (images) that satisfy the boundary conditions
• This method is particularly useful when dealing with simple geometries, such as half-spaces, wedges, or spheres, and when the boundary conditions are of the Dirichlet or Neumann type

Principle of method of images

• The main idea behind the method of images is to replace the actual problem with an equivalent problem in an unbounded domain by introducing image sources or sinks
• The image sources or sinks are placed in such a way that they satisfy the boundary conditions of the original problem
• The solution to the equivalent problem in the unbounded domain is then the sum of the contributions from the actual source and the image sources or sinks

Examples of method of images

• A point charge near an infinite grounded conducting plane can be solved using a single image charge of opposite sign placed symmetrically on the other side of the plane
• A point charge between two infinite grounded conducting planes can be solved using an infinite series of image charges placed symmetrically on both sides of the planes
• A point charge inside a grounded conducting sphere can be solved using a single image charge placed at the inverse point with respect to the sphere's surface

Limitations of method of images

• The method of images is limited to simple geometries and boundary conditions (Dirichlet or Neumann)
• It becomes increasingly complex when dealing with multiple boundaries or more complicated geometries
• The method of images is not applicable when the boundary conditions are of the mixed type or when the boundaries are not perfect conductors or insulators

Numerical methods

• Numerical methods are computational techniques for solving Poisson's equation when analytical solutions are not available or are too complex to obtain
• These methods discretize the domain into a grid or mesh and approximate the derivatives in Poisson's equation using finite differences or finite elements
• The three main classes of numerical methods for solving Poisson's equation are finite difference methods, finite element methods, and boundary element methods

Finite difference methods

• Finite difference methods approximate the derivatives in Poisson's equation using finite differences based on the values of the potential function at neighboring grid points
• The domain is discretized into a structured grid, and the Laplace operator is replaced by a finite difference approximation, leading to a system of linear equations
• Examples of finite difference methods include the central difference scheme, the Gauss-Seidel method, and the successive over-relaxation (SOR) method

Finite element methods

• Finite element methods (FEM) discretize the domain into a set of simpler subdomains, called finite elements, and approximate the solution using a linear combination of basis functions defined on these elements
• The weak form of Poisson's equation is obtained by multiplying the equation by a test function and integrating over the domain, leading to a system of linear equations
• FEM is particularly useful for solving Poisson's equation on complex geometries and with mixed boundary conditions

Boundary element methods

• Boundary element methods (BEM) reformulate Poisson's equation as an integral equation defined on the boundary of the domain, reducing the dimensionality of the problem by one
• The boundary is discretized into a set of elements, and the solution is expressed in terms of the Green's function and the boundary values of the potential function and its normal derivative
• BEM is advantageous when the domain is unbounded or when the solution is only required on the boundary

Applications of Poisson's equation

• Poisson's equation has numerous applications in various fields of physics and engineering, where it is used to model phenomena involving potential fields in the presence of sources or sinks
• Some of the main areas of application include electrostatics, gravitation, fluid dynamics, and heat transfer

In electrostatics

• In electrostatics, Poisson's equation relates the electric potential to the charge density
• It is used to calculate the electric potential and electric field distribution for given charge configurations
• Examples include determining the potential around charged conductors, dielectrics, and in plasma physics

In gravitation

• In Newtonian gravitation, Poisson's equation relates the gravitational potential to the mass density
• It is used to calculate the gravitational potential and gravitational field for given mass distributions
• Examples include modeling the gravitational field of planets, stars, and galaxies

In fluid dynamics

• In fluid dynamics, Poisson's equation arises when dealing with incompressible flows and relating the pressure to the velocity field
• The pressure field is obtained by solving Poisson's equation with the divergence of the velocity field as the source term
• Examples include modeling the pressure distribution in laminar and turbulent flows, as well as in groundwater flow

In heat transfer

• In heat transfer, Poisson's equation describes the steady-state temperature distribution in the presence of heat sources or sinks
• The temperature field is obtained by solving Poisson's equation with the heat source density as the source term
• Examples include modeling the temperature distribution in heat-generating devices, such as electronic components or nuclear reactors

Relation to other equations

• Poisson's equation is closely related to several other important partial differential equations in mathematical physics
• These equations can be seen as special cases or generalizations of Poisson's equation, depending on the nature of the source term and the presence of additional terms

Laplace's equation

• Laplace's equation is a special case of Poisson's equation when the source term is zero
• It describes the behavior of harmonic functions, which are functions that satisfy $\nabla^2 \phi = 0$
• Laplace's equation is used to model potential fields in the absence of sources or sinks, such as in electrostatics (charge-free regions), gravitation (outside mass distributions), and steady-state heat transfer (without heat sources)

Helmholtz equation

• The Helmholtz equation is a generalization of Poisson's equation that includes a linear term in the potential function
• It has the form $\nabla^2 \phi + k^2 \phi = f$, where $k$ is a constant (wave number) and $f$ is the source term
• The Helmholtz equation arises in wave propagation problems, such as in acoustics, electromagnetics, and quantum mechanics (time-independent Schrödinger equation)

Schrödinger equation

• The time-independent Schrödinger equation is a quantum mechanical analog of Poisson's equation, describing the behavior of the wavefunction in the presence of a potential energy
• It has the form $-\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi$, where $\hbar$ is the reduced Planck's constant, $m$ is the mass of the particle, $V$ is the potential energy, $E$ is the total energy, and $\psi$ is the wavefunction
• The time-independent Schrödinger equation reduces to Poisson's equation in the classical limit, when the potential energy is much larger than the kinetic energy term