Describe magnetic flux through an arbitrary area using magnetic field, area vector, surface orientation, and surface integrals.
A wire is connected to a resistor of resistance R to form a rigid rectangular loop of width L and height 2 L. An external force is exerted on the loop so that the loop always moves with constant speed v in the +x-direction, as shown in Figure 1. The loop then enters Region 1 of external uniform magnetic field of magnitude B that is directed in the -z-direction. Region 1 has boundaries x=L and x=2.5 L. The loop later enters Region 2 with two external, uniform magnetic fields, each of magnitude B, that are parallel but are directed in opposite z-directions. Region 2 has boundaries x=2.5 L and x=3.5 L. Point S is the midpoint of the leading edge of the loop and is aligned with the horizontal boundary in Region 2 that separates the two magnetic fields.
On the following axes, sketch a graph of the magnetic flux through the rectangular loop as a function of the position x of Point S from x=0 to x=4.5 L. The +z-direction indicated in Figure 1 corresponds to .
