I recently started learning the basic forms (integrals) of the Maxwell's equations, and everything that is related to electromagnetism seems to be derived from these fundamental equations. Now my question is, where did these equations come from and could you derive them (without using the strange inverted delta sign, yes, im not that far in mathematics yet) using basic calculus? Are these equations based on principles of conservation?
[Physics] Where do Maxwell’s equations come from
electromagnetismhistorymaxwell-equations
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An electromagnetic wave with a well-defined frequency and direction, i.e. $\vec k$, only has two possible truly physical i.e. transverse polarizations, i.e. the linearly polarized waves in the $x$ and $y$ direction (or the two circularly polarized ones). That implies that a truly physical counting of polarizations gives you 2, more generally $D-2$ in $D$ spacetime dimensions.
Starting from the $A_\mu$ potential fields, one component is unphysical because it's pure gauge, $A_\mu\sim\partial_\mu\lambda$, and one of them is forbidden due to Gauss' constraint $\rm div\,\vec D=0$ etc. that already constrains the allowed initial state of the electromagnetic field. Both of these killed polarizations are ultimately linked to the $U(1)$ gauge symmetry.
If one is allowed to count off-shell and unphysical fields, there may be many more components than two. But it's always possible to deduce that there are two physical polarizations at the end. For example, when we view $\vec B,\vec E$ as basic fields, there are six components, a lot. But these fields only enter Maxwell's equations through first derivatives, and not second as expected for "normal" bosonic fields, so these fields are simultaneously the canonical momenta for themselves. This brings us to three polarizations but one of them is killed by the constraints, the Maxwell's equations that don't contain time derivatives.
The Hertz vector is just the most famous "non-standard" example how to write the electromagnetic field as a combination of derivatives of some other fields. One must understand that the room for mathematical redefinitions etc. is unlimited and it is a matter of pure maths. All these descriptions may describe the same physics. At the end, the only "truly invariant because measurable" number of "fields" that all these approaches must agree about is the number of linearly independent physical polarizations of a wave/photon with a given $\vec k$.
If you can analyze any mathematical formulation of electromagnetism or another field theory and derive that there are $D-2$ physical polarizations (this usually boils down to the difference of the number of a priori fields minus the number of independent constraints and the number of parameters defining identifications i.e. gauge symmetries – but the independence is sometimes hard to see and requires you to make many steps of the counting), then you have proved everything that is "really forced to be true". Various formalisms may offer you other ways to count the number of off-shell fields (with different answers) and they may be useful (because they satisfy certain conditions or enter some laws) but to discuss them, one has to know what the laws where they enter actually are.
A truly physical approach is only one that counts the physical polarizations. The gauge symmetry is just a redundancy, a mathematical trick to get the right theory with 2 physical polarizations out of a greater number of fields with certain extra constraints or identifications. The precise number of constraints or identifications may depend on the chosen mathematical formalism and it is not a physically meaningful question – it is a question of a subjectively preferred mathematical formalism because the physics is equivalent for all of them.
- The speed of EM waves is a consequence of Maxwell equations alone. However, they do not impose constraints individually but as a collective. They let you derive a wave equation which contains the (phase) velocity as a parameter.
- Electrodynamics (as described by Maxwell's equations) is what we call a covariant theory, i.e. it is in compliance with special relativity. E.g. when you have a static charge density and you switch to a moving frame, there will also be a current density due to the moving charge density. This is exactly the same as in relativistic mechanics where time and position mix in the Lorentz-transform. In fact, the transformation is the same. There is even a ('covariant') way of rewriting Maxwell's equations such that they won't change form under Lorentz-transforms.
- Historically, electrodynamics was what motivated Einstein to pursue the idea of having the Lorentz-transform govern mechanics as well. Indeed, the original paper in which he proposed special relativity was titled 'Über die Elektrodynamik bewegter Körper' ('On the electrodynamics of moving bodies'). So in a way, no extra work was required on electrodynamics to make it relativity-ready. It was Newtonian mechanics which was flawed and needed to be fixed up by Einstein.
- It is relatively easy to show that the Lorentz-transform is as it is when you assume that the speed of light is the upper bound on velocity. It is also possible to show that given that there is an upper bound on velocity, it has to be the speed of light, but it's more difficult. I think it's hard to make an accurate statement on the importance of Maxwell's equations here. The impossibility of breaking the speed of light is a consequence of the Lorentz-transform which is motivated from electrodynamics. But it needed the genius of Albert Einstein to realize that you could also apply the Lorentz-transform on mechanics which you have to when you want to make a statement on moving bodies and their velocities.
- As to the history of the Lorentz transform, I know that it was know before Einstein published his theory of special relativity. (That's why it's Lorentz transform after Hendrik Antoon Lorentz, not Einstein-transform). But people didn't realize that is was the 'true' nature of space and time. Some thought it was an effect due to the motion of the aether, but that has been disproven experimentally by Michaelson & Morely.
Hope that helps a little.
Best Answer
The answer lies in another wiki article,
Go to the link to find the original links for each of these empirical laws.
Maxwell's tying up the various laws gathered from experimental observations and the verification of this most inclusive formulation with new observations and measurements established the theory of classical electrodynamics on par with classical mechanics. thermodynamics and statistical mechanics . It was a beautiful set of theories .
Lord Kelvin was supposed to say
(The link has a caveat that no solid reference has been found though.)
It was a belief held before the advent of x-rays , the photoelectric effect and black body radiation discrepancies.
It does set the frame for the unexpected appearance of data that led to the quantum mechanical revolution.