Illustration 29.2: Loop in a Changing Magnetic Field
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A wire loop in an external magnetic field can have an induced emf (and therefore an induced current) if the magnetic flux varies as a function of time. Since the magnetic flux is the dot product of the magnetic field and the perpendicular area of the loop (B · A for uniform magnetic fields), the flux can change if the magnitude of the magnetic field changes in time and/or if the orientation between the magnetic field and the perpendicular area changes with time (position is given in meters, magnetic field strength is given in millitesla, emf is given in millivolts, and time is given in seconds). The color of the vector indicates field strength, and graphs on the right show the magnetic field in the x direction as well as the induced emf. Restart.
In Animation 1 the loop of wire is perpendicular to the screen, while the magnetic field is to the right. The orientation of the loop and field does not change in time. However, the magnetic field strength changes with time according to the values you select with the slider. You can change the maximum magnitude of the magnetic field as well as the frequency of its oscillation.
In Animation 2 the loop of wire is again perpendicular to the screen, and now the magnetic field rotates in the plane of the computer screen. The orientation between the loop and field does change in time because the magnetic field changes direction (with respect to the loop) as a function of time. The magnetic field strength in this animation does not change with time. You can set the magnitude of the magnetic field and the frequency of the field's rotation by using the sliders.
What are the differences between the two animations? What are the similarities?
In Animation 1 the magnetic field changes strength as a function of time. In Animation 2 the magnetic field maintains a constant magnitude, but its direction changes with time. Despite these differences, for the same values of max |B| and frequency, you get the same value for the magnetic field in the x direction as a function of time and the same induced emf. For Animation 1 the magnetic field changes strength as a function of time according to sin(2π f t). In Animation 2 the magnetic field maintains a constant magnitude, but its direction changes with time. The component of the field that is in the direction of the area of the loop (a direction normal to the loop or in the x direction) changes as a function of time according to sin(2π f t). As a consequence, B · A as a function of time is the same for both animations, as long as you have the same values of max |B| and frequency.
Note that while B changes with time in one animation and changes direction in the other animation, we can still use B · A since the magnetic field at every instant in time is uniform across the area of the loop. If it were not uniform over the area of the loop, we would need to use an integral to determine the magnetic flux.
Illustration authored by Anne J. Cox and Mario Belloni.