A solenoid with length $l=25\mathrm{.}0cm,$ radius $a=2\mathrm{.}50cm$ and $42$ turns of wire carries a current that varies with time, $i(t)=(1000\frac{A}{s^2})t^2.$ Concentric with the solenoid is a single loop of wire with resistance $R=300$ and radius $b=5\mathrm{.}00cm.$ You can assume the magnetic field inside the solenoid is uniform and the magnetic field outside is negligibly small.

a$.$ Find the magnetic field inside the solenoid $B(t).$

b$.$ Find the induced current in the loop as a function of time describing its direction. Calculate the value of the induced current at $t=3\mathrm{.}00s$ and $r>a.$ Sketch a plot $ofEvsr..$

c$.$ Find the induced electric field due to the solenoid fo r and r$>$a$.$ Sketch a plot of E vs r$.$