$5(15$ points$)$

An infinitely long straight wire carries a current $\backslash ($ I $\backslash )$ in the $\backslash ($ z $\backslash )$ direction as shown in the figure below.

$1)$ Use Ampere's law to calculate the magnetic induction $\backslash ($ B $\backslash )$ as a function of radius $\backslash ($ r $\backslash )$ away from the wire.

$2)$ A square wire loop of side length $\backslash ($ a $\backslash )$ is placed on the $\backslash ($ y z $\backslash )$ plane as shown in the figure. What the magnetic flux through the loop?

$3)$ If the current on the straight wire varies linearly with time, namely $\backslash (\{\}\_\{2\}$ I$=$g t $\backslash ),$ where $\backslash ($ g $\backslash )$ is a constant and the resistance of the loop is $\backslash ($ R $\backslash ).$ What is the induced current on the loop?