Confusion in Maxwell's derivation of Ampere's Force Law

In summary, the conversation is about a question regarding Maxwell's "A Treatise on Electricity and Magnetism" and specifically about "Ampere's Force Law". The question is divided into two parts, with the first part asking about the outcome of a special case and the second part asking for a derivation of equation 21 from equation 15. The conversation also mentions the use of LaTeX on the site and provides a link for an introduction to it.
  • #1
faheemahmed6000
18
0
Hi everyone here. I have my question in the following attached pdf file
 

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  • #2
Sorry I didn't posted my question directly. It was because of unavailability of Maths symbols
 
  • #3
I doubt anyone else will read your file.
This site supports ##\LaTeX## ...
enclose your expression (say) \xi\nu with opening double-# and closing double-# to produce ##\xi\nu##
 
  • #5
I am reading Maxwell's "a treatise on electricity and magnetism, Volume 2, page 156" about "Ampere's Force Law". I have some confusion in the following pages:

Ampere2.PNG
Ampere3.PNG


My question is of two parts:
1.
Equation 20, i.e. ##P=\dfrac{B+C}{2r}## is the outcome of special case (i.e. l=1, m=0, n=0)

But in Page 156, Article 517, Maxwell says: "We can now eliminate P, and find the general value of ##\dfrac{dX}{ds}## and uses this formula (i.e. ##P=\dfrac{B+C}{2r}##) in the general case.

However in the general case, where 0 < l, m, n < 1, and hence
##\dfrac{d^{2}X}{dsds'}=l\left( \frac{dP}{ds'}\xi^{2}-\dfrac{dQ}{ds'}+(B+C)\dfrac{l'\xi}{r}\right) +m(...)+n(...)\neq0##
(since direction of X is not in the direction of ds)

therefore,
##\dfrac{dX}{ds}=l\left[ (P\xi^{2}-Q)_{(s',0)}-\int\limits_0^s' (2Pr-B-C)\dfrac{l'\xi}{r}ds'\right] +m\int\limits_0^s'(...)ds'+n\int\limits_0^s'(...)ds'##
Now in this general case, how can we get ##P=\dfrac{B+C}{2r}##.

If ##P\neq\dfrac{B+C}{2r}## in general case, what does Maxwell mean by "We can now eliminate P, and find the general value of ##\dfrac{dX}{ds}##"
2. How can one get equation 21 from equation 15. Please give a lengthy derivation.
 
  • #6
Please somebody answer my question.
 

1. What is Ampere's Force Law?

Ampere's Force Law is a fundamental equation in electromagnetism that describes the force between two electrically charged particles. It states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

2. What is the confusion in Maxwell's derivation of Ampere's Force Law?

The confusion in Maxwell's derivation of Ampere's Force Law lies in the fact that there are two different versions of the equation, one using the displacement current and one without it. This has led to debates and discussions about which version is the correct one.

3. How did Maxwell originally derive Ampere's Force Law?

Maxwell originally derived Ampere's Force Law by using the displacement current term in his equations. This term, which represents the current that is created by changing electric fields, is now included in the modern version of the equation.

4. What is the significance of the displacement current in Ampere's Force Law?

The displacement current is significant because it allows for the conservation of charge and energy in electromagnetic systems. Without this term, the equations of electromagnetism would not be consistent with the laws of conservation.

5. How has the confusion in Maxwell's derivation of Ampere's Force Law been resolved?

The confusion in Maxwell's derivation of Ampere's Force Law has been resolved by recognizing that both versions of the equation are correct in different circumstances. The version with the displacement current is needed in cases where there are time-varying electric fields, while the version without it is sufficient for static electric fields.

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