Intuitive Guide to Maxwell's Equations
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Intuitive Guide to Maxwell's Equations
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Re: Intuitive Guide to Maxwell's Equations
#2For a truly intuitive understanding I think you will have to understand relativistic differential geometry. Once you have that you just have 1 (4-dimensional) vector potential, the Laplacian of which is equal to the current.
This single remaining equation corresponds to the Maxwell equations for the electric field, the equations for the magnetic field just correspond to the fact that the 'curl' of this vector potential has 0 divergence (which is just a basic fact of geometry, and is also why magnetic monopoles are unlikely).
Re: Intuitive Guide to Maxwell's Equations
#3These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift.
On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide like this, or 3-Blue-1-Brown videos would have made my day, then.
Re: Intuitive Guide to Maxwell's Equations
#4Link to 3Blue1Brown video on Divergence and Curl: https://youtu.be/rB83DpBJQsE
Re: Intuitive Guide to Maxwell's Equations
#5As a side note there was a historical debate, and the shape of these equations is the end result of the chosen system.
In geometric algebra which did not prevail, these are one equation.
Re: Intuitive Guide to Maxwell's Equations
#6James Clark Maxwell is one of the Demi-gods of Colour science. He produced the first Colour photograph, and was the first to quantify Colour. Every time you define an RGB value, he is sitting on your shoulder.
Re: Intuitive Guide to Maxwell's Equations
#7These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…
Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.
Re: Intuitive Guide to Maxwell's Equations
#8For a truly intuitive understanding I think you will have to understand relativistic differential geometry. Once you have that you just have 1 (4-dimensional) vector potential, the Laplacian of which is equal to the current. This single remaining equation corresponds to the Maxwell equations for the electric field, the equations for the magnetic field just correspond to the fact that the 'curl' of this vector potenti…
How do you define curl in 4d?
Re: Intuitive Guide to Maxwell's Equations
#9As a side note there was a historical debate, and the shape of these equations is the end result of the chosen system. In geometric algebra which did not prevail, these are one equation.
Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations:
dF=J
dF=0
Geometric algebra produces two equations too, by the way, not one.
Re: Intuitive Guide to Maxwell's Equations
#10These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…
I love explanations of concepts that appeal to people in new and different ways, and I thank anyone clever, patient, and gifted enough to produce them. The world is a better place when more people understand are not afraid of more things.
But I think it's important to remember that mathematics is not some evil thing! There are people who train carefully so that the terse mathematical explanations are the intuition. There's nothing wrong with not being one of those people, but knocking the mathematical explanation is denying them their intuition just as their explanation denies you yours.
In a word, it takes all kinds, and I think we should all be as patient with others' ways of learning and understanding as we hope that they will be with ours.