$($adapted from Prob. $2,$ p$.399,$ Brown's Introductory Physics II$)$ Consider the diagram below. A long straight wire carries an oscillating current $I(t)=I_{0}sin(t),$ where $I_0$ is the amplitude and $$ is the angular frequency of the oscillation, and $I_0$ and $$ are constants. A rectangular loop of wire with resistance $R,$ height $b,$ and width $c$ is a distance a away from the long straight wire as shown.

$($a$)$ Using the definition of flux, find the flux through the loop due to the wire as a function of time. NOTE: The flux integral is a spatial integral, so any time dependence that shows up in the integrand will be unaffected by the integral. In other words, if $I(t)$ shows up inside the integral, it can be pulled out since it is a constant with respect to position $($even though it is not a constant with respect to time$).$

$($b$)$ Find the mutual inductance of the wire and the loop.

$($c$)$ Find the induced voltage in the loop as a function of time.

$($d$)$ Find the induced current in the loop as a function of time.

$($e$)$ Find the force between the loop and the wire as a function of time.