Normal View | Flux View

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A wire loop that is rotated by an external motor (or turbine) is shown. The rotating loop of wire is also in a constant magnetic field (created by magnets not shown). There is a current induced in the rotating loop. As the loop rotates around, you see the red (front) side and then the black (back) side of the loop (position is given in centimeters, magnetic field strength is given in tesla, emf is given in millivolts, and time is given in seconds). The green arrow indicates both the direction and magnitude of the induced current. Restart.

Consider the Normal View. The top graph shows A cos(θ), the area of the loop times cos(θ), as a function of time, where θ is the angle between the area of the loop and the magnetic field. The bottom graph shows the induced emf in the loop as a function of time.

What is the position of the loop when the magnitude of A cos(θ) is a maximum? How about when the magnitude is a minimum? What is the induced emf in the loop? Notice that when the loop is out of the screen (you see only a thin rectangle), A cos(θ) is a maximum in magnitude. When the loop points to the left in this animation, cos(θ) = 1, and when the loop points to the right in the animation, cos(θ) = - 1. When the loop is completely in the plane of the screen, then cos(θ) = 0. Notice that the induced emf is related to the negative of the slope of the A cos(θ) vs. time graph. Why?

Now consider the Flux View in which the graphs on the right show the flux through the loop and the induced emf in the loop as a function of time. What is the position of the loop when the magnitude of the magnetic flux is a maximum? When is the magnitude of the magnetic flux a minimum?

Notice that the flux is the dot product between B and A or just BA cos(θ) for uniform magnetic fields (magnetic fields that are uniform across the area of the loop). If the magnetic field is not uniform we must use an integral. Therefore, when A cos(θ) is a maximum [cos(θ) = 1] or a minimum [cos(θ) = -1], so is the flux. Similarly, when A cos(θ) is zero, so is the flux. Notice that the corresponding induced emf is related to the negative of the slope of the magnetic flux vs. time graph. Since the A cos(θ) vs. time graph is proportional to the magnetic flux vs. time graph, with the proportionality constant being the magnetic field strength, this explains the relationship between A cos(θ) and the induced emf in the Normal View.

In electric power plants turbines generate current based on this principle. Either a wire rotates in a magnetic field (as in this Illustration) or, more commonly, a magnet rotates near stationary coils of wire (changing the magnetic flux through the coils, which induces current in them). For electricity in the United States, turbines make 60 revolutions per second (generating 60 Hz current), while in Europe the turbines make 50 revolutions per second (generating 50 Hz current).

Illustration authored by Melissa Dancy, Anne J. Cox, and Mario Belloni.
Script authored by Wolfgang Christian, Melissa Dancy and Anne J. Cox.