Physics Asked on December 17, 2020
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?
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.
Answered by user4552 on December 17, 2020
When you get the E and B components from the four potential you take the gradient of phi and time derivative of A and the curl of A. Linear dependence/independence may not be preserved across derivatives. This might be the reason why you get additional linear independences in the EM field.
Example: Consider the function f(x) = x^2. Now take a derivative of f w.r.t. x and x^2. You get 2x and 1 respectively. Before the derivative you had one linearly independent component, but after the derivative you have two.
Answered by Sap on December 17, 2020
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