Electromotive force and current in a superconducting solenoid with limited length induced by a bar magnet and a monopole

The magnetic flux $\Phi_{\mathrm{B}}$, electromotive force, EMF, and current $I_{\mathrm{in}}$, induced by a moving magnetic bar and an imaginary magnetic monopole in a superconducting solenoid of multiple turns and length $L$, are numerically calculated. The magnetic field of the bar magnet is approximated with the magnetic field along $z$ axis of a solenoid with length $l$ and radius $a$ and current $I$, while the magnetic field of the monopole is supposed to be inversely proportional to $r^{\mathrm{2}}$. Calculations show that, for a bar magnet, $\Phi_{\mathrm{B}}$ and $I_{\mathrm{in}}$ essentially saturate when the bar moves inside superconducting solenoid, so EMF is zero while $I_{\mathrm{in}}$ is constant. EMF is only induced when the bar enters and exits the solenoid and $I_{\mathrm{in}}$ is zero after the bar leaves the solenoid. For a magnetic monopole, $\Phi_{\mathrm{B}}$ is discontinuous (from positive maximum to negative maximum) when the it moves through each turn of the superconducting solenoid, but EMF caused by $d\Phi_{\mathrm{B}}$/\textit{dt} is continuous while the EMF induced by the a moving monopole is a delta function (moving monopole produces a ring-shaped $E$ field). The total EMF$_{\mathrm{Tot}}$ in solenoid is the superposition of EMF of each turn of coil and the plateau appears. The current $I_{\mathrm{in}}$ continues to grow while the monopole leaves the solenoid.