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See here for an introduction: https://www.physicsforums.com/help/latexhelp/robphy said:This site supports LATEX\LaTeX ...
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.
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.
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.
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.
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.