Chapter 9: Problem 24
Use a Fourier transform in three dimensions to find a solution of the Poisson equation: \(\nabla^{2} \Phi(\mathbf{r})=-4 \pi \rho(\mathbf{r})\).
Short Answer
Expert verified
The solution of the given three-dimensional Poisson equation is \(\Phi(\mathbf{r})=-\frac{1}{2\pi^2}\int \frac{d^3k}{k^2}e^{i \mathbf{k}\cdot \mathbf{r}} \tilde{\rho}(\mathbf{k})\).
Step by step solution
01
Apply Fourier Transform
Apply the Fourier transform to both sides of the Poisson equation. Remember the Fourier transform of a function \(f(\mathbf{r})\) is computed as \(F(k)=\int d^3r e^{-i \mathbf{k}\cdot \mathbf{r}} f(\mathbf{r})\). The transformed Poisson equation becomes \( (2 \pi)^2 k^2 \tilde{\Phi}(k) = -4 \pi \tilde{\rho}(k). \)
02
Rearrange the expression
Re-arrange the above equation to isolate \(\tilde{\Phi}(k)\) on one side of the equation. We obtain the expression \( \tilde{\Phi}(k)=-\frac{2\pi}{k^2} \tilde{\rho}(k) \). This gives \(\tilde{\Phi}(k)\), the Fourier transform of the potential, in terms of \(\tilde{\rho}(k)\), the Fourier transform of the charge density.
03
Inverse Fourier Transform
The final step is to apply the inverse Fourier transform to this equation and find the potential \(\Phi(\mathbf{r})\). Compute the inverse Fourier transform as \(f(\mathbf{r})=\int d^3k e^{i \mathbf{k}\cdot \mathbf{r}} F(\mathbf{k})\). If you apply the inverse Fourier transform, you find the final solution to the given equation which is \( \Phi(\mathbf{r})=-\frac{1}{2\pi^2}\int \frac{d^3k}{k^2}e^{i \mathbf{k}\cdot \mathbf{r}} \tilde{\rho}(\mathbf{k})\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fourier Transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, much like a musical chord can be expressed as the frequencies of its individual notes. In physics and engineering, this transform is a powerful tool for solving differential equations, particularly those that are more easily understood in the frequency domain than in the time or spatial domains.
For a function \(f(\mathbf{r})\), where \(\mathbf{r}\) represents a position vector in three-dimensional space, the Fourier transform is defined as \(F(k) = \int d^3r e^{-i \mathbf{k}\cdot \mathbf{r}} f(\mathbf{r})\). The variable \(k\) stands for the wavevector, which relates to the frequency of the component sine waves. The exponent involving \(i\), the imaginary unit, signifies that this is a complex transform, capturing both amplitude and phase information of \(f(\mathbf{r})\) at different frequencies. Using this tool, we can transform a spatial description of a system into a frequency description, which is often more manageable analytically.
For a function \(f(\mathbf{r})\), where \(\mathbf{r}\) represents a position vector in three-dimensional space, the Fourier transform is defined as \(F(k) = \int d^3r e^{-i \mathbf{k}\cdot \mathbf{r}} f(\mathbf{r})\). The variable \(k\) stands for the wavevector, which relates to the frequency of the component sine waves. The exponent involving \(i\), the imaginary unit, signifies that this is a complex transform, capturing both amplitude and phase information of \(f(\mathbf{r})\) at different frequencies. Using this tool, we can transform a spatial description of a system into a frequency description, which is often more manageable analytically.
Mathematical Physics
Mathematical Physics encompasses the application of mathematics to problems in physics and the development of mathematical methods for such applications and for the formulation of physical theories. The Poisson equation, used in our exercise, is one such example where mathematical concepts directly apply to physical phenomena. It is a partial differential equation of elliptic type, which in this context, relates the electric potential \(\Phi(\mathbf{r})\) to the charge density \(\rho(\mathbf{r})\) within a field.
In the realm of electrostatics, the Poisson equation is instrumental in describing the potential field generated by a given charge distribution. The equation \(abla^2 \Phi(\mathbf{r}) = -4 \pi \rho(\mathbf{r})\) is quite challenging to solve in the spatial domain because of the complexity of differential operators. However, leveraging the Fourier transform simplifies the process, converting the differential equation into an algebraic equation that is far easier to manipulate. Mathematical Physics often involves such transformations and manipulations to extract meaningful solutions from complex problems.
In the realm of electrostatics, the Poisson equation is instrumental in describing the potential field generated by a given charge distribution. The equation \(abla^2 \Phi(\mathbf{r}) = -4 \pi \rho(\mathbf{r})\) is quite challenging to solve in the spatial domain because of the complexity of differential operators. However, leveraging the Fourier transform simplifies the process, converting the differential equation into an algebraic equation that is far easier to manipulate. Mathematical Physics often involves such transformations and manipulations to extract meaningful solutions from complex problems.
Charge Density
Charge density, denoted as \(\rho(\mathbf{r})\), represents the amount of electric charge per unit volume at a point in space. It is a crucial concept in electrostatics and directly ties into the Poisson equation. In physical terms, this describes how charge is distributed throughout a particular region.
Understanding the charge density is essential for predicting the electromagnetic fields and forces in a system. The exercise given employs the concept of charge density as the source term in the Poisson equation. In mathematical terms, a higher charge density at a point corresponds to a stronger influence on the potential field at that point. Moreover, the charge density can vary significantly within different media, and across boundaries, which can substantially affect the resulting electric potential. Through the Fourier transform solution method, we relate the complexities of the spatially varied \(\rho(\mathbf{r})\) to its simpler frequency domain counterpart, \(\tilde{\rho}(\mathbf{k})\), which is used to find the potential distribution.
Understanding the charge density is essential for predicting the electromagnetic fields and forces in a system. The exercise given employs the concept of charge density as the source term in the Poisson equation. In mathematical terms, a higher charge density at a point corresponds to a stronger influence on the potential field at that point. Moreover, the charge density can vary significantly within different media, and across boundaries, which can substantially affect the resulting electric potential. Through the Fourier transform solution method, we relate the complexities of the spatially varied \(\rho(\mathbf{r})\) to its simpler frequency domain counterpart, \(\tilde{\rho}(\mathbf{k})\), which is used to find the potential distribution.
Inverse Fourier Transform
The inverse Fourier transform is the process of reconstructing a function from its frequency domain representation back into its original form, typically time or space domain. In our exercise, after transforming the Poisson equation and finding the potential in the frequency domain \(\tilde{\Phi}(k)\), we need to convert it back to the spatial domain to obtain the actual electric potential \(\Phi(\mathbf{r})\) felt at every point in space.
The inverse Fourier transform is given by \(f(\mathbf{r}) = \int d^3k e^{i \mathbf{k}\cdot \mathbf{r}} F(\mathbf{k})\), which reconstitutes the spatial function from its frequency components. This transformation is symmetrical to the Fourier transform but with a positive sign in the exponent and denotes an integral over the wavevectors. In the case of the Poisson equation, this step is critical to move from the algebraic solution back to a physical one that describes the potential in real space, completing the process of solving the Poisson equation using Fourier analysis.
The inverse Fourier transform is given by \(f(\mathbf{r}) = \int d^3k e^{i \mathbf{k}\cdot \mathbf{r}} F(\mathbf{k})\), which reconstitutes the spatial function from its frequency components. This transformation is symmetrical to the Fourier transform but with a positive sign in the exponent and denotes an integral over the wavevectors. In the case of the Poisson equation, this step is critical to move from the algebraic solution back to a physical one that describes the potential in real space, completing the process of solving the Poisson equation using Fourier analysis.