Ciro Santilli
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Maxwell's equations | 🗖 nosplit | ↑ parent "Particle physics" | 428, 12, 954

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Unified all previous electro-magnetism theories into one equation.
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Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.
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The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.
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TODO: what does it mean that it is coherent with special relativity?
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The equations are a system of partial differential equation.
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The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:
• , , : directions of the electric field
• , , : directions of the magnetic field
and two known input functions:
• : density of charges in space
• : current vector in space. This represents the strength of moving charges in space.
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Due to the conservation of charge however, those input functions have the following restriction: $$Equation 1. Charge conservation. ∂t∂ρ​+∇⋅J=0 (1)$$
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Also consider the following cases:
• if a spherical charge is moving, then this of course means that is changing with time, and at the same time that a current exists
• in an ideal infinite cylindrical wire however, we can have constant in the wire, but there can still be a current because those charges are moving
Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.
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The goal of finding and is that those fields allow us to determine the force that gets applied to a charge via the Equation 6. "Lorentz force".
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and then to find the force we just need to integrate over the entire body.
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Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations: $$Equation 2. Gauss' law divE=ε0​ρ​ (2)$$
🔗$$Equation 3. Gauss's law for magnetism divB=0 (3)$$
🔗$$Equation 4. Faraday's law of induction ∇×E=−∂t∂B​ (4)$$
🔗$$Equation 5. Ampere's circuital law ∇×B=μ0​(J+ε0​∂t∂E​) (5)$$
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and you should review the interpretation of divergence and curl :
• divergence: how much fluid goes out or into a point
• curl: points in the direction in which a wind spinner spins fastest
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