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Illustration 29.1: Varying Field and Varying Area


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In this chapter we consider Faraday's law, which tells us how a changing magnetic flux creates an electromotive force (an emf), -dΦ/dt = emf. Magnetic flux, Φ, is a measure of the amount of magnetic field flowing perpendicularly through an area. It is given by B · A for uniform magnetic field and constant area (position is given in meters, magnetic field strength is given in milliTesla, emf is given in millivolts, and time is given in seconds). Restart.

Consider the Changing Magnetic Field Animation. A loop is shown in a region in which the magnetic field varies sinusoidally, then is constant, and finally starts to vary sinusoidally again. The graphs on the right show the induced emf in the loop and the magnetic flux through the loop as a function of time. The direction of current in the top of the loop is indicated by the current arrow above the loop. Blue indicates the magnetic field is into the page; red indicates it is out of the page. The intensity of the color is proportional to the magnitude of the magnetic field.

Notice that, for the first 1.5 s of the animation, there is an increasing flux through the loop due to the magnetic field that is increasing out of the screen. Also notice that there is an emf induced in the wire loop and an induced clockwise current. Is the induced current in the loop in the direction you would expect? The induced current may be in the opposite direction as you expected. Because of the minus sign in Faraday's law (Lenz's law), the emf is the negative of the slope of the flux vs. time graph. From t = 0 s to t = 1.5 s the magnetic field is increasing and therefore the emf is negative. Now watch the animation for the remaining time and see how the emf changes with time.

Consider the Changing Area Animation. A loop of changing area is shown in a region where the magnetic field is constant (unlike the previous animation where the magnetic field changed with time) and pointing out of the screen. The graphs on the right again show the induced emf in the loop and the magnetic flux through the loop as a function of time. The direction of current in the top of the loop is indicated by the current arrow above the loop. Blue indicates the magnetic field is into the page; red indicates it is out of the page.

Note that (again) the magnetic flux is increasing for the first 1.5 s of the animation. Again note that the emf is negative during this time interval. In fact, compare the two graphs from the first animation with the pair of graphs from the second animation. What do you notice? Since the emf is related to the changing flux, it does not matter if that changing flux is due to a changing magnetic field or a changing area. In fact, a changing magnetic flux can be due to a changing magnetic field, a changing area, or both.

Illustration authored by Melissa Dancy and Mario Belloni.

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