# Nature of Faraday-Induced Electric Fields

I'm having some trouble with Griffiths's explanation of induced electric fields. I know a Faraday-induced electric field will arise when there's a varying magnetic field (equivalently, a variation in its flux). The author says that in order for us to build a sort of correspondence with the laws of magnetostatics we need to consider the analog between $$-\frac{\partial \mathbf{B}}{\partial t}$$ and $$\mu_0 \mathbf{J}$$. Thus, we can obtain Faraday's law in its integral form (Ampère's law in magnetostatics): $$\oint_{\gamma} \mathbf{B} \cdot d\mathbf{l} = \mu_0I_{enc} \Rightarrow \oint_{\gamma} \mathbf{E}_{ind} \cdot d\mathbf{l} = -\frac{d \Phi_{\Sigma}}{dt}$$ where $$\gamma \equiv \partial \Sigma$$ is the boundary of Amperian loop.
I'm not quite sure I understand this last equation: What I think the author is trying to convey is that in order for us to determine the direction of the Faraday-induced electric field, we basically need to cleverly exploit the fact the combination of the universal flux rule $$+$$ Lenz's law tells us how the induced current runs in a loop immersed in a varying $$\mathbf{B}$$.
This means that if we place a fictitious Amperian loop in the right way we are able to determine what electric field has caused the moving charges in the loop to start moving and generating the induced current. Now the actual question is:
Why would an induced electric field arise if there's no loop at all? Nature abhors a change in flux and this is okay, but without a surface to begin with, magnetic flux is not a mathematically definable quantity. How would nature even know? The strategy of placing an imaginary Amperian loop lies in the fact we can now safely apply all those rules and laws we obtained (induced emfs, induced currents,...). Is a Faraday-induced field just a way for Nature to take (allow me to say this in the humblest of manners) a precautionary action? Any insight would be much appreciated.

• Faraday's law is a generalization of experience with Faraday's experiments. It can't be derived or guessed based on the Ampere law. Commented Aug 29, 2023 at 16:41

The field $$\mathbf E$$ is induced irrespective of whether there is a loop in your mind in said field or not. Faraday's law states that the integral of any electric field along any closed integration contour equals the negative rate of magnetic flux change enclosed by that very loop. That is all. There is no causality here, that is one side of the equation causes the other, there is no current mentioned, only that the equality of a contour (line) integral with that of the rate of a surface integral. If the flux rate is zero, for example, then the closed loop integral is zero implying a conservative static field, as it should.