I was learning about proof of Neumann's law on electromagnetic induction. In that proof, they have considered a circular loop carrying current i, a small current element dl and magnetic induction $\overrightarrow B$. Now, I have learnt that only variable current can produce inductive emf or magnetic induction. Now, as I followed the proof step by step, While calculating the total work done, they have placed i (current) outside of integration. Now, it is so obvious that they are treating i as a constant. But again, how can a constant current produce magnetic induction?
Here is some initial steps of proof(in book):
Let I be the length of a conductor which is part of a closed circuit shown in the figure . Let $\overrightarrow B$ be the magnetic induction acting at any point on an elementary part $\overrightarrow {dl}$ of the conductor. If i is the current through $\overrightarrow {dl}$ due to the emf $E$ of a cell included in the closed circuit, then the force on $\overrightarrow{dl}$ is $i \overrightarrow{dl} X \overrightarrow B$. If due to this force, the element $\overrightarrow {dl}$ has elementary displacement $\overrightarrow{dx}$ in time $dt$, the work done by the external source (i.e., the cell) on the element is $i (\overrightarrow{dl}X \overrightarrow B).\overrightarrow{dx}.$
The whole conductor is free to move, the total work done is,
$\\[10pt]i\int(\overrightarrow{dl}X \overrightarrow{B})\; \overrightarrow{dx}=i\int(\overrightarrow{dx}X \overrightarrow{dl}).\;\overrightarrow B$
This is the part where I am getting confused (where i is outside the integral). Can anyone please clarify? Also I have another doubt related to the angle $\theta$ between $\overrightarrow{dl}$ and $\overrightarrow{B}$. If the conducting loop is experiencing some force and causing displacement of the position of the wire, then will angle or $\theta$ vary continuously ?(in that way will $sin \;\theta$ also be outside the integral ?
Please help me clarify these two doubts related to $i(current)$ and angle $\theta$.
