$3.$ Solenoid S whose cross$-$section is shown in Fig. $3$ is wrapped with $\backslash ($ n $\backslash )$ turns of wire per its unit length. It also contains a thin insulating circular loop L of radius $\backslash ($ r $\backslash )$ that is coaxial with the solenoid. A small positively charged bead B $($given charge $\backslash ($ q $\backslash )$ and mass $\backslash ($ m $\backslash ))$ can slide without friction along this loop. As observed now, the counterclockwise current feeding the solenoid wire equals to $\backslash ($ I$\_\{1\}\backslash ),$ while the bead moves with speed $\backslash ($ u$=$u$\_\{1\}\backslash )$ in the clockwise direction. Then the feeding current is increased in some fashion to magnitude $\backslash ($ I$\_\{2\}>$I$\_\{1\}\backslash ).$ Would the bead's speed change as a result $($and why$)?$ If not, then state that $\backslash ($ u$\_\{2\}=$u$\_\{1\}\backslash ).$ Otherwise, find the "new" speed $\backslash ($ u$\_\{2\}\backslash )$ after the current becomes equal to $\backslash ($ I$\_\{2\}\backslash ).$ Given that the particle can move $($that$\text{'}$s$,$ velocity$)$ in two opposite directions on the loop, we refer to speed here as the magnitude of the velocity $\backslash ($ v: u$=|$v$|\backslash ).$

Figure $3$: Illustration for problem $3.$