I know that Faraday's law gives a recipe for finding the EMF over a closed loop. But, other than through symmetry arguments, I have not found a way discussed in my textbooks on how to find explicit functional form of the non-conservative field. Would there be a basic law which we can use the to find the non-conservative electric field of general set up? Similar to how we can integrate coulombs law to find electrostatic fields of general set ups.
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asked Sep 22, 2021 at 1:45
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1$\begingroup$ What about the answer "Maxwell's equations" doesn't sit well with you? Do you have a specific system in mind? $\endgroup$– BioPhysicistCommented Sep 22, 2021 at 2:08
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$\begingroup$ I am not very familiar with how to solve PDES especially ones involving the vector operators.. so I was checking if there was some more elementary recipe one could use $\endgroup$– Cathartic EncephalopathyCommented Sep 22, 2021 at 2:09
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Consider Faraday's Law in Differential Form: $$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} $$ This is basically a set of 3 coupled partial differential equations, which could be solved to get the non-conservative electric field.
But for Generality we could in theory find the electric field by inverting the curl operator: (provided we know the magnetic field $\vec{B}$. $$ \vec{E} = (\nabla \times)^{-1} -\frac{\partial \vec{B}}{\partial t}$$ Check this paper to find the closed inverse operator: https://arxiv.org/pdf/0804.2239.pdf
So yeah, there is a sort of general methodology for finding the non-conservative Electric Field, but that being said, I feel it would be much more easier to figure out the Magnetic Vector Potential $\vec{A}$ from the current densities and then know that the non-conservative component of the Electric field becomes: $$\vec{E}_{nc} = -\frac{\partial A}{\partial t}$$
P.S Feel like this question came from a person who is just taking his first courses in Electrodynamics in which case none of what I said above may make any sense. Just know that the laws of electromagnetism that you study will be rewritten more elegantly in the form of Maxwells Equation which when done will give you differential equations relating the Electric, Magnetic fields to currents and charges. And you could solve these equations to find the Electric and Magnetic Fields. (easier said than done)
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$\begingroup$ Certainly, I have tried to read griffiths but I didn't do further than the polarized material chapter. I'll choose the best answer after seeing the other answers which maybe posted. Very cool that you can invert curl btw $\endgroup$ Commented Sep 22, 2021 at 2:29
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2$\begingroup$ @Buraian To be clear, you cannot "invert the curl." If you're handed a divergenceless field $\vec Y$ and asked to find some $\vec X$ such that $\nabla \times \vec X = \vec Y$, then (assuming your domain is $\mathbb R^3$) you can do so; however, this $\vec X$ will be highly non-unique. Furthermore, when it comes to E&M even this procedure only makes sense if you've been handed $\frac{\partial\vec B}{\partial t}$, but in general you need to solve for $\vec E$ and $\vec B$ simultaneously. In other words, Faraday's law only provides a constraint that $\vec E$ and $\vec B$ must satisfy. $\endgroup$ Commented Sep 22, 2021 at 3:20