Solving Maxwell's Equations: A Challenge

In summary, the problem is that the author is not understanding how to calculate the divergence of electric fields generated by a point charge. Classical electromagnetism deals with singularities by assuming that point charges are the wrong way to look at what is really going on.
  • #1
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Hello all... I have been working on this problem that I just am not being able to solve.

I've been spending my spare time learning some vector calculus and non-euclidean geometry (my aim is to be able to finally tackle relativity). After learning some basic things about the del function, I found that I had sufficient mathematical knowledge to be able to derive Maxwell's equations (well I thought I did).

I had a go at the first of the four. The way I am trying to do it is by taking Coulomb's law, writing it as a vector in three dimensions and then taking the divergence. I am hoping to get this from it:

[tex]\nabla{\mathbf{.E}} = \frac{\rho}{\epsilon}[/tex]

But I keep on getting 0. And I have no idea why...
Here is a scanned copy of my working, I would be very grateful if you could point out my errors (other than the minus sign maybe) to me.

Cheers.

http://postimage.org/image/q44ym4ro/ [Broken]
 
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  • #2
For the hypothetical situation you described, with a single point charge at (0,0,0), the charge density should be zero everywhere except at (0,0,0).

The charge density would be q times a 3 dimensional (generalized) Dirac delta. This way the volume integral of [itex]\rho[/itex] for a volume containing (0,0,0) would be equal to q.

Perhaps it would be simpler to just use the integral forms when dealing with point charges.
 
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  • #3
If you mean taking divergence of E from point charge, you have to use Cauchy Formula, in which case you will get exactly q δ(x)/ε = ρ/ε.
 
  • #4
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Definition of divergence.

If you assume the charge has a continuous distribution over space, the divergence won't be 0. So if you put q = ρV in Coulomb's law, where ρ is the charge density and V is the volume of some arbitrary region, you'll be able to derive the equation. The volume V must be a function of x, y, and z of course.
 
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  • #5
So, the problem lies in not the mathematics but my interpretation of it?

Why does a point charge yield a zero divergence but a charge enclosed in finite volume (say a sphere of radius R) yield finite divergence when the electric fields generated by them at point satisfying [tex]x^2 + y^2 + z^2 \geq R^2 [/tex] is the same. An imaginary sphere around the point charge has field lines leaving the volume enclosed through the surface, and the field lines generated by those two configurations of charge are the same at a large distance.
 
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  • #6
Also, is there no other way to deal with it (other than using dirac delta function)? Does this mean that there is a singularity there? How does classical electromagnetism deal with this singularity? Or are the equations inadequate (meaning that point charges are the wrong way to look at what is really going on)?
 
  • #7
If you misinterpret the [itex]\delta[/itex] distribution as a function, there's a singularity, but it's not a function but a functional on the space of sufficiently smooth and sufficiently quickly falling test functions, and that's the adequate description of a point particle in a field theory.
 

1. What are Maxwell's Equations?

Maxwell's Equations are a set of four equations that describe the relationship between electric and magnetic fields, as well as their interactions with charged particles. They are fundamental to the study of electromagnetism and have numerous applications in modern technology.

2. Why is solving Maxwell's Equations considered a challenge?

Solving Maxwell's Equations involves complex mathematical calculations and requires a deep understanding of electromagnetic theory. Additionally, the equations are often coupled, meaning that solving one requires knowledge of the others, making it a challenging task.

3. What are the main applications of solving Maxwell's Equations?

Maxwell's Equations are used in a wide range of applications, including the design of electronic circuits, the development of communication technologies, and the study of light and optics. They are also essential in the development of advanced technologies such as wireless power transmission and superconductivity.

4. How have Maxwell's Equations contributed to our understanding of the universe?

Maxwell's Equations have played a crucial role in our understanding of the universe. They have helped us understand the nature of light and electromagnetism, as well as the behavior of particles at the atomic and subatomic levels. They also form the basis of Einstein's theory of relativity, which revolutionized our understanding of space and time.

5. What are some common methods used to solve Maxwell's Equations?

There are several methods used to solve Maxwell's Equations, including analytical methods, numerical methods, and computer simulations. Each method has its advantages and limitations, and the choice of method depends on the specific problem being solved. Some commonly used techniques include finite element analysis, finite difference time-domain methods, and method of moments.

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