electromagnetism - Maxwell-Faraday's Law of Induction Violated?

I am having much distress over Maxwell's 3rd Equation (Faraday's Law of Induction) and a thought experiment I had. Given that Maxwell-Faraday's equation is $$\oint E \cdot ds = -\frac{d\phi}{dt}$$And from the definition by HyperPhysics (emphasis mine),

The line integral of the electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area enclosed by the loop.

If this is the case, please consider the following scenario.

I insert a dense magnetic field into ONLY THE CENTER of a loop of wire (the magnetic field does not touch the actual loop). I was taught that Faraday's Law of Induction could be derived from the Lorentz Force on moving charges exposed to magnetic fields. However, as no magnetic field interacts with the charges in the wire (the field doesn't extend to the coil) there should be no EMF induced. But Maxwell's equations says there should be because there is a change in flux in the area of the loop.

I'm pretty sure Maxwell's equations aren't wrong, so could someone please explain what's wrong here? Does Maxwell's equation assume that the flux change is uniform through the entire area? That doesn't sound like an assumption that he would make, given the universality of his 4 equations.

Answer

Your assumption that there has to be a magnetic field interacting with the wire is wrong. It is not only the magnetic field what moves the charges, it is also an electric field. Inside the area in which the magnetic field is changing the rotor of the electric field is not zero. This creates a contour condition for the electric field, which results in a non zero value for it outside the region, even if the magnetic field is zero there. The magnetic part of the Lorentz Force on moving charges is only one of the components of the EMF, see https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction#Proof