Discrepancies with Maxwell's Eqns - vector potentials

In summary: This usually involves looking up the symbols and their meanings in a table, like in the article linked. Then I can use that as my starting point and work from there. It's good to have a few conventions to work with, so you're not constantly flipping back and forth between different conventions.
  • #1
VictorVictor5
13
0
Greetings all,

Trying to resolve a discrepancy with vector and scalar potentials with Maxwell's Equations, specifically Ampere's law.

In my E&M textbook (Balanis, 1989, Eqn 6-17), Ampere's law with a magnetic vector potential and electric scalar potential can be expressed as

[tex]E= -\nabla\phi-j \omega A [/tex]

where [tex] \phi [/tex] is the electric scalar potential, and A is the magnetic vector potential.

Now, in a paper I am referencing in my work, I see Ampere's expressed as the following:

[tex] E=-j \omega(A- \nabla \phi) [/tex]

When you distribute this equation, you get the [tex] -j \omega A + j \omega \nabla \phi[/tex]

where now the scalar potential is positive, and also has a [tex] j \omega [/tex] in front of it, where the first equation doesn't.

Is it because of the scalar potential being arbitrary since it's a function of position? Or is there something else?

I also checked Harrington, but no luck there either.

Thanks!
VV5
 
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  • #2
My guess is the notation in the paper, compared to the notation you're used to is related by [tex]\phi = -\frac{\phi}{jw}[/tex]

So it's just a unit conversion basically. You see this a lot in EM. There's several different conventions.
 
  • #3
DuckAmuck,

First, thanks for the reply.

Question for you. While the ratio you provided would work, and given that -j * -j = well, j^2, = -1 and the equation would work, but quick question. The ratio you provided - physically what would that mean?

Thanks again!
VV5
 
  • #4
VictorVictor5 said:
DuckAmuck,

First, thanks for the reply.

Question for you. While the ratio you provided would work, and given that -j * -j = well, j^2, = -1 and the equation would work, but quick question. The ratio you provided - physically what would that mean?

Thanks again!
VV5

It's just a unit convention. For example, in particle physics we like to set speed of light equal to 1, for simplicity, so we're not writing "c" over and over. So E=mc^2 becomes E=m. The consequence of this is that energy and mass are in the same units, which is okay, just something to keep in mind when doing problems.

It doesn't really mean much *physically*, it's just a writing convention. And actually, the convention I'm used to doesn't even use j, all components are real. :)
 
  • #5
Great - thanks so much for your help!

VV5
 
  • #6
VictorVictor5 said:
Great - thanks so much for your help!

VV5

What I generally try to do is work through what the paper does in the convention I'm used to.
 

1. What are Maxwell's equations and what is their significance in physics?

Maxwell's equations are a set of fundamental equations that describe the behavior of electric and magnetic fields. They are significant because they provide a unified framework for understanding and predicting the behavior of electromagnetic phenomena, such as light, electricity, and magnetism.

2. What are discrepancies with Maxwell's equations and why are they important?

Discrepancies with Maxwell's equations refer to situations where the equations do not accurately describe the behavior of electromagnetic fields. These discrepancies are important because they can lead to new discoveries and insights into the nature of electromagnetic interactions, and may also help to improve our understanding of fundamental physical principles.

3. What is a vector potential and how does it relate to Maxwell's equations?

A vector potential is a mathematical construct used in electromagnetism to describe the behavior of electric and magnetic fields. It is related to Maxwell's equations through the use of the vector calculus operator known as the curl, which allows for the calculation of the electric and magnetic fields from the vector potential.

4. How do discrepancies with Maxwell's equations impact our understanding of electromagnetism?

Discrepancies with Maxwell's equations can challenge our current understanding of electromagnetism and may lead to the development of new theories and models to explain these discrepancies. They also highlight the limitations of our current understanding and push for further research and experimentation to better understand the behavior of electromagnetic fields.

5. What research is being done to address discrepancies with Maxwell's equations?

There is ongoing research in the field of electromagnetism to address discrepancies with Maxwell's equations. This includes developing new theories and models, as well as conducting experiments to test and validate these theories. Additionally, advancements in technology and mathematical techniques are also being explored to better understand and address discrepancies in Maxwell's equations.

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