A $56\mathrm{.}42-$cm$-$diameter wire loop lies flat in the plane of the page. The $16-$gauge copper wire used to make the loop has a diameter of $1\mathrm{.}29$ mm $.$ The positive $n-$axis $(n$ for 'normal'$)$ points out of the page and the center of the loop is at the origin. The magnetic field passing through the page is

vec$(B)=[0\mathrm{.}20(T)+(0\mathrm{.}10\frac{(T)}{(s)})t-(0\mathrm{.}050\frac{(T)}{s^2})t^2]hat(n).$

$($a$)$ Sketch a graph of the magnetic flux versus time through the loop for $0t4\mathrm{.}0s.$

$($b$)$ You are looking down on the loop from the $+n-$axis. Determine the magnitude and direction of the current in the loop at $t=0\mathrm{.}5s.$ The resistivity of copper is $1\mathrm{.}77{10}^{-8}*m.[$Hint: You will need to review the relationship between resistivity and resistance.$]$

$($c$)$ Repeat part $($c$)$ for $t=2\mathrm{.}0s.$

$($d$)$ Repeat part $($c$)$ for $t=4\mathrm{.}0s.[$Extra Credit: Using the graph in part $($a$)$ and your results from $($b$)-($d$),$ make a general statement about when the current changes direction.$]$

$($e$)$ Use Faraday's Law to determine the magnitude of the induced electric field in the loop at $t=0\mathrm{.}5$ s $.$