OpenStax College Physics for AP® Courses, Chapter 23, Problem 8 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. A circular coil lies in the plane of the page— I have drawn it in black here— and it has an area of 0.250 square meters and it's going to be stretched so this end is going to be pulled out and this end will be pulled out and so that these sides go together until it has no area at all because it will just be a straight line and that happens in 0.100 seconds. It's in the presence of a magnetic field, which is straight into the page of 1.50 tesla and there are 50 turns of wire in this coil so the question is what emf will be induced in the coil? Because the flux through the coil is going to change as a result of the reduction in area and so there will be an induced current that creates a magnetic field that attempts to oppose the change and so... you know, the question isn't explicitly asking this but the magnetic field direction that's induced would be into the page in order to replace the flux that's going to be taken away as the area reduces. Okay! So emf is the number of loops multiplied by the rate of change in flux and so that is by the change in flux divided by the amount of time it takes for that change to happen. Flux is the magnetic field strength multiplied by the area of the loop times cosine of the angle between the perpendicular to the loop and the magnetic field. So a perpendicular to the loop goes straight in and out of the page and so it's parallel to the magnetic field, in other words, and so the angle between them is zero so we are gonna have cosine of zero, which is 1; this is the orientation where you have the maximum flux. Okay! So we plug this in here and this is the change in flux because it starts... this is the initial flux and then the final flux is zero and so the change is all of this initial flux. The negative sign just reminds us that the direction opposes the change in flux but that's not important in this question. So the emf then is 50 turns times 1.50 tesla times 0.250 square meters—original area— times cos of 0 divided by 0.100 seconds and that's 187.5 volts. So the induced emf is 188 volts, in other words. Well actually it does ask for the direction of the emf (sorry about that) so all this stuff I have been saying about the induced magnetic field direction is important. So yeah the magnetic field induced will be into the page to replace the flux that's going to be taken away and in order to do that, it will have to have a current that's going in this direction. It's going to be... we can figure this out by grabbing the loop, let's grab the bottom of the loop here say with our right hand and point our fingers into the page and our thumb will point in the direction of current and so it's going to be going clockwise, when seen from above. So there we go: 188 volts, clockwise from above will be the induced emf.