EMF Generated according to Faraday's Law

Question

According to Faraday's Law, due to a relative movement between the current carrying loop and the magnetic field, an EMF is induced in the loop causing a current flow. However, according to Maxwell-Faraday's Law, $$\nabla \times \bf E = -\frac{\partial \bf B}{\partial t}$$ Clearly, if the magnetic field changes, the electric field becomes non-conservative and electric potential is no longer defined. In that case, how can we even define the emf of the circuit? What does induced emf actually mean then?

Answer 1

how can we even define the emf of the circuit?

The EMF is defined as a line integral of the electric field.

According to Faraday's Law, due to a relative movement between the current carrying loop and the magnetic field, an EMF is induced in the loop

Notice the wording: The EMF is induced on (or around) the loop, not "between two points" as we would say about a potential difference.

Of course we can also have an EMF for an open path between two points (as opposed to a loop), but we must specify the path to be taken between the points. The EMF between the points can be different for different paths between the points.

Clearly, if the magnetic field changes, the electric field becomes non-conservative

Again, this is exactly it. If the field were conservative, then the EMF around any loop in that field would be 0. Only with a non-conservative field can we have a non-zero EMF around a closed loop.

Since we define the EMF for each path of integration between two points, the EMF will be well-defined even when the potential difference between two points in a system is not.