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In a setup like the one show in the figure:

enter image description here

Will there be an induced EMF across the loop(given that the field is increasing)?

(my dilemma is : since the loop is not complete, we cannot define the flux through it.)

Is it valid to define Flux for an incomplete loop?

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    $\begingroup$ Intuitively I expect the magnetic field lines to be squeezed a tiny bit into the dielectric gap of the capacitor and some zealous finite element field analysis might confirm but in the real world it is unlikely to be detectable. The loop is closed by a reactive impedance if the magnetic field is changing. $\endgroup$
    – KalleMP
    Commented Aug 28, 2023 at 6:33

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Yes.

What will happen is that an emf will induced in a complete loop which consists of a part of the complete loop shown with full lines and another part of the complete loop defined by a dotted line (indicating that it is not a conductor) between the plates of the capacitor.

The electric field due to the induced emf will then move mobile charge carriers so that the capacitor becomes charged until such a time as the electric field due the charges on the plates of the capacitor, is equal in magnitude but opposite in direction to, the induced electric field produced by the changing magnetic flux through the loop.
Then the mobile charge carriers will cease to move because the net force on them due to the two opposing electric fields is zero.

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