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Shaun Culver
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Can Maxwell's equations (the usual 4 equations) be summarized in the form of one equation?
kurt.physics said:Is not that the relativistic maxwell equation? Hey, there should be a mock up t-shirt for that
I'll try to answer this. Part of the problem in giving the full answer is giving proper historical context... I will do my best in a summary.patfla said:Was it (is it) the case that Maxwell's original equations were not spacetime (or frame-of-reference) invariant?
That is, two different frames of reference with the same observable and Maxwell's original equations would yield different results.
And while I don't recognize *d*F(A) = J, it's my understanding that maybe the original purpose of the Lorentz transform was to fix this problem?
Hmm... just to be clear, Maxwell's equations today are the same as Maxwell's "original" equations (although we have found more convenient ways of writing them, for example the equations usually seen in undergrad textbooks are how Heavyside rewrote them with vector notation).patfla said:Yes that is: if one is trying the apply Maxwell's original equations from different coordinate systems. I.e. you would get different results (on the same observable). I would assume that at that time, the 'fundamental' inertial frame was space (or the aether) while the moving frame-of-reference (with a different coordinate system) was the Earth.
Oh yes, there was all kinds of confusion. Check out the Michelson-Morley experiment, Trouton-Noble experiment, Trouton-Rankine experiment, just to name a few. What is interesting to point out is that the last experiment listed there was after Einstein's special relativity paper. It took quite awhile for people to adjust to these new ideas.patfla said:Did contradictions actually arise (I assume they must have)?
Hmm... just to be clear, Maxwell's equations today are the same as Maxwell's "original" equations (although we have found more convenient ways of writing them, for example the equations usually seen in undergrad textbooks are how Heavyside rewrote them with vector notation).
f-h said:Actually F(A) is a two-form...
roughly: the hodge star turns an n-form into a 4-n form. So we start with a 1-form A take the exterior derivative: dA (2-form) -> *dA (2-form) -> d*dA (3-form) -> *d*dA (1-form)
the hodge star is defined such that w wedge *v = <w,v> vol where <w,v> is a sort of inner product turning the n-forms into a function and vol is the volume form of the metric. Since the volume form is an m-form in m-dimensional spacetimes in you can see that the hodge star needs to turn the n-form v into an (m-n)-form *v such that w wedge *v can be the volume m-form.
shaunculver said:Can Maxwell's equations (the usual 4 equations) be summarized in the form of one equation?
Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They were developed by James Clerk Maxwell in the 19th century and are a cornerstone of classical electromagnetism.
Yes, they can be summarized into a single equation known as the Maxwell's equation of electromagnetism. This equation combines the four original equations into one, making it easier to understand and use in various applications.
Maxwell's equations are important because they form the basis of classical electromagnetism, which is essential for understanding and predicting the behavior of electric and magnetic fields in various systems. They have also been crucial in the development of modern technologies such as electric power, telecommunications, and electronics.
Yes, Maxwell's equations are still relevant and widely used in various fields such as physics, engineering, and telecommunications. They have been extensively tested and found to accurately describe the behavior of electric and magnetic fields, making them a fundamental part of modern science and technology.
Yes, Maxwell's equations have been found to have applications beyond classical electromagnetism. They have been used in other areas of science, such as optics, acoustics, and even in the theory of relativity. This shows the versatility and importance of these equations in understanding the natural world.