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I'm taking a course on electrodynamics and I'm confused when we start talking about potentials. The electromagnetic field seems to have 6 independent components to me. It's described by six dynamical equations (and 2 non-dynamical constraint equations). The EM field tensor has 6 independent components as well.

However, when describing the field in terms of the EM four-potential, the EM field seems to be completely determined by four independent components. What am I missing here?

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/20071/2451 , physics.stackexchange.com/q/249397/2451 and links therein. $\endgroup$ – Qmechanic Nov 17 '19 at 23:54
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    $\begingroup$ @Qmechanic Those questions are not at all helpful. The answer to the questions you linked is that the EM field is not overdetermined since two of Maxwell's equations are non-dynamical. That's a given in the question I asked. $\endgroup$ – Eric Nov 18 '19 at 0:06
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I'm not really sure what this question is asking that is different from what the other two asked, but possibly the following is what you want. Consider the following, simpler example. In Newtonian mechanics, the gravitational field is the gradient of the potential. So at a point, there are 3 degrees of freedom, because the field is a vector. But if you want to choose how the field varies in a neighborhood, you can't make the variation be however you like. E.g., you can't give it a nonzero curl. Therefore the 3 d.f. at nearby points are in some sense not independent of each other.

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    $\begingroup$ What I'm still confused about is this: The EM field tensor contains 6 linearly independent components but the EM four potential contains only 3 independent components after taking into account the gauge freedom. So my question: why do we need 6 independent functions of space and time in order to determine the electromagnetic field when 3 such functions should actually suffice? $\endgroup$ – Eric Nov 18 '19 at 3:16

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