When the magnet is moved very far away, how much flux is inside the ring compared to the original flux ${\mathit{\Phi }}_{\mathbf{0}}$? How much of this flux is due to the current in the ring?

Electric flux is the rate at which an electric field flows through a given area. Electric flux is directly proportional to the number of electric field lines passing through the virtual surface.

Consider a bar magnet and the conducting ring is shown in the following figure.

The magnetic field is induced within the ring when a current is passing through it. Give magnetic flux in the ring is${\Phi }_0$ . After a bar magnet is moving far away from the ring, the magnetic flux remains the same as${\Phi }_0$ .

If you move the magnet away from the ring, then the flux passing through the ring decreases.

Thus, the induced current is developed in the ting. This induced current develops induced magnetic flux in the same amount, which opposes the change in the origin flux. Hence, the net magnetic flux remains the same as the original flux.