Separation of Variables In Electrostatics

In summary, the speaker is questioning the legitimacy and accuracy of using Separation of Variables in problem-solving, specifically in the context of Electromagnetism. They are concerned about removing portions of the solution in spherical problems and how this may affect the overall understanding of the solution. Additionally, they are unsure about the reliability of finding constants through Fourier Series and if the constant found at one specific point can be applied to the entire space being considered. However, the speaker acknowledges that using sines and cosines to construct solutions may provide some justification. They are open to further explanation and clarification on these concepts.
  • #1
M-Speezy
15
0
I am curious how legitimate a solution Separation of Variables tends to give. I've been working problems out of Griffith's book on Electromagnetism, and am often uneasy as to the way things are done. I have two specific issues. The first, is that in spherical it is often necessary to remove entire portions of the solution, as you would divide by zero otherwise. This makes perfect sense at surface level, because otherwise the solution wouldn't make any sense. But how do you know that is then the solution, that you haven't left off important parts? The other issue I have is with finding the constants through a Fourier Series. Often, the potential would be given at a specific point, so everything is simplified to be analyzed at JUST that specific point, so that the constant can be found. My question is how do we know that the constant found at that point will apply to all space (that is being considered)? It all 'makes sense', but seems very dubious to me. If anyone would be able to explain some of these ideas, or maybe why I'm perhaps being silly, I would really appreciate it. Thanks!
 
Physics news on Phys.org
  • #2
I think I may have answered one of my questions, the one about ignoring portions of solutions. Griffiths discusses the fact that sines or cosines can be used to construct any solution, so it is somehow justifiable. I think to understand it at a deeper level I'd be looking at a lot of math. My other question still stands, if anyone knows anything about it!
 

Related to Separation of Variables In Electrostatics

1. What is separation of variables in electrostatics?

Separation of variables in electrostatics is a mathematical technique used to solve problems involving the distribution of electric charges and the resulting electric fields. It involves breaking down a complex problem into simpler, independent parts that can be solved separately and then combined to find a solution to the original problem.

2. How does separation of variables work in electrostatics?

In electrostatics, separation of variables works by assuming that the electric potential or electric field can be expressed as a product of two or more simpler functions, each depending on only one variable. These functions are then substituted into the governing equations of electrostatics, such as Gauss's law or Poisson's equation, to obtain a system of ordinary differential equations that can be solved using standard mathematical techniques.

3. What are the advantages of using separation of variables in electrostatics?

One advantage of using separation of variables in electrostatics is that it allows for the simplification of complex problems, making them easier to solve. It also allows for the use of well-known mathematical techniques, such as solving ordinary differential equations, which can save time and effort compared to other methods. Additionally, separation of variables can provide insight into the behavior of electric fields and charges in a system.

4. What are some common applications of separation of variables in electrostatics?

Separation of variables is commonly used in electrostatics to solve problems involving charged particles or conductors, as well as problems involving electric fields in different geometries. It is also used in the study of electrostatic potential and capacitance in circuits, as well as in the analysis of dielectric materials and their effects on electric fields.

5. Are there any limitations to using separation of variables in electrostatics?

While separation of variables is a powerful technique in electrostatics, it does have some limitations. It is most effective for solving problems with linear boundary conditions and uniform charge distributions. It may also be difficult to apply in situations with complex geometries or non-linear boundary conditions. In these cases, other numerical or analytical methods may be more suitable.

Similar threads

I An attempt to understand the associated variable conventions in circuits
    • Electromagnetism
Replies
1
Views
860
Formal difference between electrostatics and magnetostatics
    • Electromagnetism
Replies
6
Views
3K
I Work done by electric field of an irregularly shaped conductor
    • Electromagnetism
Replies
3
Views
603
I Determining equilibrium solutions to differential equations
    • Differential Equations
Replies
16
Views
964
Two hemispheres, and a shell inside dielectric material
    • Electromagnetism
Replies
1
Views
891
I Statistical modeling and relationship between random variables
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
564
I Electrostatic force exerted on an electron inside a nucleus
    • Classical Physics
Replies
19
Views
890
B Projected linear separation between companion stars
    • Astronomy and Astrophysics
Replies
14
Views
2K
Show uniqueness in Dirichlet problem in unit disk
    • Calculus and Beyond Homework Help
Replies
6
Views
459
I Visual Interpretation of Advanced Electrodynamics
    • Electromagnetism
Replies
1
Views
722

Hot Threads

Recent Insights

Back
Top