When there is a change in magnetic-flux, based on Faraday's law of induction & Lenz's law, we know that there is change in Potential Difference now, aside from the source V now we have a induced −V due to the change in magnetic-flux, and it opposes the current, why would it? I understood from lenz's law that it will, but not great detail as to why.
This comes from Maxwell's equations. Start with Faraday's Law:
$$ \nabla \times E = -\frac{\partial B}{\partial t}$$
Note the negative sign. If we integrate both sides by the area through which there is a magnetic field (think of the cross-section of a solenoid, for example), then we get:
$$ \int (\nabla \times E) \cdot dA = \oint E\cdot dl = - \frac{\partial \Phi}{\partial t}$$
The middle term follows from applying Stoke's theorem on the first term. The middle term is also the definition of voltage. The final definition is flux ($\Phi$) which is the product of the magnetic field and the cross-sectional area the field passes through, which is how one arrives at the third term.
Another thing, if the power-source can be increased, V can potentially increase to oppose the −V? And maintain A at the same value it was?
The 'opposite V' or back EMF acts to stabilize the current flowing. Suppose you increase V, and hence A, which increases the flux and hence slowly increases the back EMF. Then suppose that you stop increasing V. The back EMF will persist and slowly decay thereafter until the current stops changing (current reaches steady-state). If you were to instead keep increasing V, you'd keep increasing A, keep increasing the flux and keep increasing the back EMF (it would exacerbate the situation).
