5.8.1: Illustrations

Illustration 1: Varying Field and Varying Area

In this chapter we consider Faraday's law, which tells us how a changing magnetic flux creates an electromotive force (an emf), \(-d\phi /dt =\text{ emf}\). Magnetic flux, \(\phi\), is a measure of the amount of magnetic field flowing perpendicularly through an area. It is given by \(\mathbf{B}\cdot\mathbf{A}\) 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 \(1.5\text{ s}\) 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 \(t = 0\text{ s}\) to \(t = 1.5\text{ s}\) 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 \(1.5\text{ s}\) 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.

Illustration 2: Loop in a Changing Magnetic Field

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 (\(\mathbf{B}\cdot\mathbf{A}\) 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 \(\sin(2\pi f t)\). 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 \(\sin(2\pi f t)\). As a consequence, \(\mathbf{B}\cdot\mathbf{A}\) 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 \(\mathbf{B}\cdot\mathbf{A}\) 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.

Illustration 3: Electric Generator

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 \(A \cos(\theta )\), the area of the loop times \(\cos(\theta )\), as a function of time, where \(\theta\) 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 \(A \cos(\theta )\) 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), \(A \cos(\theta )\) is a maximum in magnitude. When the loop points to the left in this animation, \(\cos(\theta ) = 1\), and when the loop points to the right in the animation, \(\cos(\theta ) = - 1\). When the loop is completely in the plane of the screen, then \(\cos(\theta ) = 0\). Notice that the induced emf is related to the negative of the slope of the \(A \cos(\theta )\) 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 \(\mathbf{B}\) and \(\mathbf{A}\) or just \(BA \cos(\theta )\) 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 \(A \cos(\theta )\) is a maximum \([\cos(\theta ) = 1]\) or a minimum \([\cos(\theta ) = -1]\), so is the flux. Similarly, when \(A \cos(\theta )\) 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 \(A \cos(\theta )\) 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 \(A \cos(\theta )\) 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 \(60\text{ Hz}\) current), while in Europe the turbines make \(50\) revolutions per second (generating \(50\text{ Hz}\) current).

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

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.