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Magnetic Field Changes as a Function of Time

A magnetic field is perpendicular to a loop. The graph shows how the magnetic field changes as a function of time, with positive values for B indicating a field into the page and negative values a field out of the page. Several points on the graph are labelled.

1. At which point or points is the magnitude of the flux a maximum?
2. At which point or points is the magnitude of the flux a minimum? c) At which point or points is the flux changing most rapidly?
3. At which point or points is the flux changing least rapidly (but non-zero)?
4. Sketch the induced current versus time. On this plot label a-e and the direction of current by the sign: define clockwise flow as positive and counter-clockwise flow to be negative.

1. The magnitude of the flux is maximum when the magnetic field is maximum and directed perpendicular to the loop. From the graph, this occurs at point B.
2. The magnitude of the flux is minimum when the magnetic field is zero or directed parallel to the loop. From the graph, this occurs at points A and D.
3. The flux changes most rapidly when the slope of the magnetic field versus time graph is the steepest. From the graph, this occurs at point C.
4.  The flux is changing least rapidly when the slope of the magnetic field versus time graph is the least steep but non-zero. From the graph, this occurs at points A and D.
5. To sketch the induced current versus time, we use Faraday’s law, which states that the magnitude of the induced emf is proportional to the rate of change of the magnetic flux through the loop. The direction of the induced current can be determined using Lenz’s law, which states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it.

At point A, the magnetic field is decreasing and directed into the page. This induces a current in the loop in the clockwise direction, as indicated by the negative sign on the graph.

At point B, the magnetic field is increasing and directed out of the page. This induces a current in the loop in the counterclockwise direction, as indicated by the positive sign on the graph.

At point C, the magnetic field is changing most rapidly, which induces the largest emf and current in the loop. The direction of the current is counterclockwise, as indicated by the positive sign on the graph.

At point D, the magnetic field is decreasing and directed out of the page. This induces a current in the loop in the clockwise direction, as indicated by the negative sign on the graph.