A varying current i(t) flows through a long, straight wire in the xy-plane as in Example 5. The current produces a magnetic field \(\mathbf{B}\) whose magnitude at a distance \(r\) from the wire is \(B=\frac{\mu_{0} i}{2 \pi r} T\), where \(\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T}-\mathrm{m} /\) A. Furthermore, \(\mathbf{B}\) points into the page at points \(P\) in the xy-plane.

Assume that \(i(t)=t(12-t) \mathrm{A}(t\) in seconds \()\). Calculate the flux \(\Phi(t)\), at time \(t\), of \(\mathbf{B}\) through a rectangle of dimensions \(L \times H=3 \times 2 \mathrm{~m}\) whose top and bottom edges are parallel to the wire and whose bottom edge is located \(d=0.5 \mathrm{~m}\) above the wire, similar to Figure 13(B). Then use Faraday's Law to determine the voltage drop around the rectangular loop (the boundary of the rectangle) at time \(t\).