Question

Properties of the Coulomb and Sereened Coulomb Potentials (a) From the integral represcntation of the dalta function, $$ \delta(\mathbf{r})=\int \frac{d \mathbf{k}}{(2 \pi)^{3}} e^{\mathbf{k} \cdot \mathbf{r}} $$ and the fact that the Coulomb potential \(\phi(r)=-e / r\) satisfies Poisson's equation, $$ -\nabla^{2} \phi(\mathbf{r})=-4 \pi e \delta(\mathbf{r}) $$

Short answer

Through the steps provided, we demonstrate that the Coulomb potential \(\phi(r) = -\frac{e}{r}\) satisfies Poisson's equation with the relation \( -abla^2 \phi(\mathbf{r}) = -4 \pi e \delta(\mathbf{r})\).

Step-by-step solution

Understand the Coulomb potential and Poisson's equation

The Coulomb potential \(\phi(r) = -\frac{e}{r}\) describes the electric potential due to a point charge. Poisson's equation \(abla^2\phi(\mathbf{r}) = -4\pi e \delta(\mathbf{r})\) relates the Laplacian of the potential field to the charge distribution, where \(\delta(\mathbf{r})\) is the Dirac delta function representing a point charge.

Substitute the Coulomb potential into Poisson's equation

Using the Coulomb potential, compute the Laplacian \(abla^2\phi(\mathbf{r})\) and substitute it into Poisson's equation to verify it satisfies the relationship.

Compute the Laplacian of the Coulomb potential

Take the Laplacian of the Coulomb potential \(abla^2\left(-\frac{e}{r}\right)\) using spherical coordinates, where \(r = \sqrt{x^2 + y^2 + z^2}\) is the radial distance from the origin.

Compare with Poisson's equation

After calculating \(abla^2\phi(\mathbf{r})\), compare the result to the right-hand side of Poisson's equation \(-4\pi e \delta(\mathbf{r})\) to confirm that the Coulomb potential satisfies the equation.

Consider the Fourier representation of the delta function

Also, consider the given integral representation of the delta function \(delta(\mathbf{r})=int \frac{d \mathbf{k}}{(2 \pi)^{3}} e^{\mathbf{k} \cdot\mathbf{r}} \)and use it to express the delta function in the context of Poisson's equation.

Key concepts

Poisson's Equation

Poisson's equation is a fundamental mathematical equation in electrostatics that describes the relationship between the electric potential \( \phi(r) \) in a region and the charge distribution within that region. In its three-dimensional form, the equation is expressed as \( -abla^2 \phi(\mathbf{r}) = -4 \pi \rho(\mathbf{r}) \) where \( \rho(\mathbf{r}) \) is the charge density and the negative sign on the right-hand side indicates the convention for the direction of electric field lines pointing from positive to negative charges.

For a point charge, the charge density is represented using the Dirac delta function, \( \delta(\mathbf{r}) \) and Poisson's equation simplifies to \( -abla^2 \phi(\mathbf{r}) = -4 \pi e \delta(\mathbf{r}) \) with \( e \) being the charge of the point charge. The delta function indicates that the entire charge is concentrated at a single point in space.

Delta Function

The delta function, also known as the Dirac delta function, is a theoretical construct that is used to model an infinitely sharp peak at the origin, with the total integral over the space equal to one. It is not a function in the traditional sense but rather a distribution, used in physics and engineering to represent an idealized point mass or point charge.

Mathematically, it is defined by its property of sifting out the value of a function at a certain point during an integral: \[ \int f(\mathbf{r}') \delta(\mathbf{r} - \mathbf{r}') d\mathbf{r}' = f(\mathbf{r}) \]. In the context of Poisson's equation, the delta function represents a source of electric potential at a singular point. The integral representation given in the exercise, \[ \delta(\mathbf{r})=\int \frac{d \mathbf{k}}{(2 \pi)^{3}} e^{\mathbf{k} \cdot \mathbf{r}} \], showcases the connection between the delta function and Fourier transforms.

Laplacian Operator

The Laplacian operator, denoted as \( abla^2 \) or \( \Delta \) is a second-order differential operator in the n-dimensional Euclidean space. It is used to measure the rate at which a function spreads out from a point. Formally, for a scalar function \( \phi(\mathbf{r}) \), the Laplacian \( abla^2\phi \) captures the divergence of the gradient of the function. In three-dimensional Cartesian coordinates, \( abla^2 \) can be written as \[ abla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \].

In electrostatics, when applied to the electric potential, the Laplacian operator relates the curvature of the potential function to the charge density, giving rise to Poisson's equation. In spherical coordinates, which are more suitable for central forces like the Coulomb potential, the Laplacian has a more complex form that accounts for the geometry of the space.

Fourier Representation

The Fourier representation of a function or a signal is a way of expressing the function as a sum of sinusoids. This representation is crucial in understanding the behavior of functions and solving differential equations, especially those with periodicity or convolution operations.

Fourier analysis decomposes functions into oscillating functions which are characterized by frequencies. This transformation from the time or space domain into the frequency domain is carried out through the Fourier transform. In the context of the delta function given in the exercise, the Fourier representation \[ \delta(\mathbf{r})=\int \frac{d \mathbf{k}}{(2 \pi)^{3}} e^{\mathbf{k} \cdot \mathbf{r}} \] is the Fourier transform of the delta function, highlighting the intrinsic connection between spatial and frequency domains. This fundamental principle is not only pivotal in signal processing but also in solving Poisson's equation in electrodynamics where charge distributions generate potential fields in space.