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Maxwell wrote it all out component-by-component, direction-by-direction, in twenty seperate equations. So the electric field in north-south direction depends upon what the magnetic field in the east-west direction is doing, for example. The equations work in three dimensions, and relate fields pointing in different directions to each other.
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That is, visible light, radio, wifi, X-rays and the rest, depending on the wavelength. Those waves are electromagnetic radiation. So not only do the fields become real physical objects - something that Faraday was the first to propose - but they can carry travelling waves. Mathematically this is expressed in the fact that the equations can be rearranged and combined to get a new kind of equation, that describes a travelling wave. A changing electric field causes a changing magnetic field, which will cause more changes in the electric field, and so on. The equations show that electric and magnetic fields can exist even in the absence of electric charges. the equations of nature often seem to know much more than the people who discover them Fields and Waves What is amazing is how much that framework then reveals, both in terms of deep physical principles, and rich physical phenomena. The equations seem to have been very much a bottom-up affair, in that Maxwell collected together a number of known laws which were used to describe various experimental results, and (with a little extra ingredient of his own) fitted them into a unified framework. There’s a letter, dated March 1865, from William Thomson (later Lord Kelvin) saying he was sorry for being slow, that he’d read most of it and it seemed pretty good (“decidely suitable for publication”). It was submitted in 1864 but, in a situation familiar to scientists everywhere, was held up in peer review. Original manuscript of Maxwell’s seminal paper Photograph: Jon Butterworth/Royal Society
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