# Recipe for finding non-conservative electric field in general case

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.

• What about the answer "Maxwell's equations" doesn't sit well with you? Do you have a specific system in mind? Sep 22, 2021 at 2:08
• 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 Sep 22, 2021 at 2:09

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: $$\vec{E}_{nc} = -\frac{\partial A}{\partial t}$$
• @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. Sep 22, 2021 at 3:20