Exploring Laplace's Equation: Separation of Variables Method in Electrodynamics

[vikasagartha]

I am studying Laplace's equation in my electrodynamics course (using griffiths intro to electrodynamics). I am watching a youtube video stepping through the separation of variables method for solving the PDE. It seems to be a common PDE that comes up repeatedly in physics (Helmholtz eqn, Poisson eqn) and I thought it would be worthwhile to understand where the solution comes from.

In the text Griffiths also touches on two additional points:
- "A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere"
- A uniqueness theorem

A first question that came to my mind: why do we assume that separation of variables works? How can we assume that the solution is of the form $f(x)g(y)$? What if it had an expression like $x^y$?

 

[SteamKing]

The functions chosen as solutions using separation of variables must be shown to be solutions of the particular PDE and to satisfy whatever boundary conditions apply. These requirements eliminate many elementary functions from consideration, because they cannot satisfy one or both of these requirements.

 

[vikasagartha]

Okay, a second question: I watched Chris Tisdell's (youtube) derivation of the solution in two dimensions, and it involves a summation of a sine function (a Fourier series of sorts). In electrostatics, the solution of the equation generates a potential function V, which we get by taking a path integral $\int \vec{E}\cdot\vec{dl}$. I've done many such problems and never see any sinusoidal functions in there...why is that? Does the solution look different in 3D or am I misunderstanding the relation between the derivation of the solution and its application to electrostatics?

 

[dauto]

What you have to understand is that the separation method gives you a complete set of solutions (there are theorems that prove that for a fairly large group of PDE problems called Sturm–Liouville equations.) What that means is that even though there are solutions that cannot be obtained by the separation of variables method, these solutions can be written as an expansion in terms of the solutions obtained by the separation of variables method. This expansion is a generalization of the Fourier expansion. So, even though the final solution may not be sinusoidal at all, it can still be expanded in terms of sine functions because the set of sine functions form a complete set according with Sturm–Liouville theory.

 

[PJC01]

I am glad to hear that you are exploring Laplace's equation in your electrodynamics course and using Griffiths' textbook as a resource. The separation of variables method is a powerful tool for solving partial differential equations (PDEs) and it is indeed a common technique used in physics, particularly in electrodynamics. It is important to understand where this method comes from and why it is applicable to Laplace's equation.

To answer your first question, the reason we assume that separation of variables works for Laplace's equation is because it is a linear PDE. This means that the solution can be written as a linear combination of simpler solutions. In this case, the simpler solutions are the functions f(x) and g(y). This assumption is based on the principle of superposition, which states that the overall solution to a linear PDE can be found by adding together simpler solutions.

Regarding your concern about the solution having an expression like x^y, this would not be considered a solution to Laplace's equation. The reason is that Laplace's equation is a second-order PDE, meaning that the highest derivative in the equation is of second order. This rules out any solutions that have higher order derivatives, like x^y. In other words, the solution must satisfy the specific form of Laplace's equation in order to be considered a valid solution.

The two additional points mentioned by Griffiths are also important to understand. The fact that the average value over a spherical surface is equal to the value at the center of the sphere is known as the mean value property of Laplace's equation. This property is useful in applications where the solution represents a physical quantity, as it tells us that the value of the solution at any point is equal to the average value over a surrounding surface.

The uniqueness theorem mentioned by Griffiths is also an important concept. It states that if a solution to Laplace's equation exists and satisfies certain boundary conditions, then it is the only possible solution. This is important because it ensures that we are finding the correct solution to the problem at hand.

In summary, the separation of variables method is a powerful tool for solving Laplace's equation and its variations. It is applicable due to the linearity of the equation and is based on the principle of superposition. It is important to understand the mean value property and uniqueness theorem associated with Laplace's equation in order to fully grasp its applications in physics. I encourage you to continue