$[4]$ A rectangular onducting loop of width $\backslash ($ a $\backslash )$ and length $\backslash ($ b $\backslash )$ is located near a long onducting wire carrying a ourrent $\backslash ($ I$\_\{1\}\backslash )$ as shown in Figure $4.$ The long side of the loop is parallel to the long wire and the distance between the wire and the closest side of the loop is $\backslash ($ x $\backslash ).$

$($a$)$ Caloulate the total magnetic flux through the rectangular loop due to the aurrent in the long wire. The magnitude of magnetic field due to the aurrent in the long wire at the distance $\backslash ($ r $\backslash )$ is $\backslash ($ B$=\backslash$frac$\{\backslash$mu$\_\{0\}$ I$\_\{1\}\}\{2\backslash$pi r$\}\backslash ).$

$($b$)$ Find the mutual inductance $\backslash ($ M $\backslash )$ of the rectangular loop with respect to the long wire.

$($c$)$ When the rectangular low is moving to the right with constant speed $\backslash ($ v $\backslash ),$ determine the induoed enf of the locp.

$($d$)$ Find the magnitude of the induced aurent of the loop and explain the direction of it$.$ The total resistance of the loop is $\backslash ($ R $\backslash ).$