5.7.1: Illustrations

Illustration 1: Fields from Wires and Loops

The magnetic field around a long, straight wire carrying current out of the computer screen points in a direction that circles the wire (position is given in meters and magnetic field strength is given in tesla). You may want to use a right-hand rule to determine the field direction. If you point the thumb of your right hand out of the computer screen (as if it pointed in the direction of the current) and close your fingers in a fist, then your fingers will point in the direction of the magnetic field around the wire. Instead of one wire, add four wires (again the current in these wires is coming out of the screen). Notice that the vectors from each wire add up. Double-click inside the animation to draw a field line.

What would you expect the direction of the field and the field lines to look like for many wires all lined up in a horizontal row? Sketch your prediction first. Once you have your prediction, try it by pushing the "plate" button. Explain why the field lines look like they do.

For the plate, you should have predicted that the fields from the individual wires that point in the y direction should all cancel. This leaves just a field in the \(x\) direction. Since the currents in the wires are all out of the screen, the field points to the left above the plate and points to the right below the plate.

Now, let's put a loop perpendicular to the page. In this representation you are looking at the edge of a loop of wire: The wire goes into the screen, circles around, and comes back out. The blue and red dots simply represent a slice of the wire, with red indicating current coming out and blue indicating current going in. For this case, describe how the current travels (Does the current flow into the screen at the top or the bottom of the loop?). The field points to the right along the center axis of the loop and diverges out from there. Adjust the size of the loop by click-dragging either the red or blue dot. Notice that the region near the center of the loop becomes more and more uniform as the loop gets bigger and bigger.

If you place many loops side by side, what do you expect? Try it by pushing the "solenoid" button. What are the similarities and differences between the field inside a solenoid and the field above a plate? Again the magnetic fields in the y direction all cancel, leaving the magnetic field in the \(x\) direction. Given that there is no current enclosed for an Amperian path in the plane with sides outside of the solenoid, the magnetic field is zero there. Inside the solenoid, however, the magnetic field is rather uniform and points to the right.

In order to use Ampere's law, you will need to have a sense of the direction of magnetic fields from a wire or group of wires and then build Amperian loops to match the symmetry of the fields.

Illustration authored by Anne J. Cox.
Script authored by Mario Belloni and Wolfgang Christian and modified by Anne J. Cox.

Illustration 2: Forces Between Wires

The wire in the center has a fixed current. You can change the current in the blue wire by using the slider (position is given in meters, current is given in amperes, and magnetic field strength is given in tesla). The animation shows the magnetic field vectors (you can also double click on the screen to draw the field lines). Restart.

Keep the current in the blue wire at zero. In what direction is the magnetic field at the point where the blue wire is located? When the current is turned on in the blue wire, the current will either come out of the page (positive) or go into the page (negative). If you put positive current in the blue wire, in what direction would the force be on those moving charges (the current in the wire)? This is the Lorentz force on the charge carriers, so you'll need to use the right-hand rule you used in the previous chapter. Use the slider to put positive current through the blue wire. The vector shown is the force on the wire. Using the right-hand rule, the direction of the force is \(q\mathbf{v}\times\mathbf{B}\). The positive charges are moving out of the screen and the direction of the magnetic field is in the plane of the screen, and perpendicular to the line separating the currents. Using the right-hand rule gives you a direction toward the red wire.

Move the blue wire to a new position. The force points in a different direction at this position, but it still points toward the other wire. What happens if you increase the current? The force gets bigger. What happens if you make the current negative? Now the direction of the current changes and so will the direction of the force from the right-hand rule. The currents will now repel instead of attract.

Why is there a force vector on the center (red) wire? Well, this wire also experiences a magnetic field due to the blue wire. We get the force on the red wire to be equal and opposite to that of the force on the blue wire from the right-hand rule, but we could have just as easily predicted this result from Newton's third law.

Illustration authored by Anne J. Cox.

Illustration 3: Ampere's Law and Symmetry

A single wire carrying current in the \(z\) direction (out of the computer screen) has radial symmetry about the center of the wire. Two systems that differ only by a rotation about the center of the wire will be indistinguishable. This symmetry is, however, broken if a second wire is added, because the displacement vector from the first to the second wire defines a unique direction. Calculations to determine magnetic field strength that depend on the ability to follow a closed Amperian path are much more difficult because it is not possible to write a simple analytic expression for a path along which \(|B|\) is constant.

Look at the magnetic field vectors for the one-wire configuration. Notice how there is a circular symmetry about the center of the wire. Because of this symmetry, we can use Ampere's law to determine the magnetic field. Now look at the configuration with two wires. You can drag the wires either toward or away from each other. Notice that with two wires the magnetic field lines no longer have a circular symmetry. As a consequence we cannot use Ampere's law to determine the magnetic field. Do not think that Ampere's law is no longer valid. Ampere's law is always true. It is just that in certain cases it is easy to use Ampere's law to calculate the magnetic field and in others it is too difficult.

What is the analytic expression for the magnetic field on a path that has constant \(|B|\) if there is only one wire? Move the wires closer together and farther apart. Under what circumstances can this expression be used as an approximation in the case of two wires? Around one wire, \(|B| =\mu_{0} I / 2\pi r\) and it points in the direction tangent to a circle centered on the wire. If there are two wires, we may add together the magnetic fields due to each individual wire. Be careful: You must add these fields as vectors, not as just numbers.

The magnetic field produced by two long, straight wires is far from irregular. What types of symmetry does this system still have? There is still a symmetry in the \(z\) direction, but this symmetry is not useful for calculations using Ampere's law. Why? How do you use an Amperian loop to calculate the magnetic field? To use this symmetry, the loop or rectangle would have to be centered on the wire and have one side along the \(z\) axis and the other side in the \(xy\) plane. Using this rectangle, how much current is enclosed? Since the loop is infinitesimally thin, there is no current enclosed, and therefore the result for the magnetic field is zero. We already know that the magnetic field in the \(z\) direction and the radial direction are zero. So this symmetry also tells us something about the field.

Exploration 4: Path Integral

Ampere's law is \(\int\mathbf{B}\cdot d\mathbf{l}=\mu_{0}I\), where the integration is over a closed loop (closed path), \(d\mathbf{l}\) is an element of the path in the direction of the path, \(\mu_{0}\) is the permeability of free space (\(4\pi\times 10^{-7}\text{ T}\cdot\text{m/A}\)), and \(I\) is the total current enclosed in the path. When you turn the integral on, this animation shows the path integral as you move the pencil around (results from the path integral are given in \(10^{-7}\text{ T}\cdot\text{m}\)). Restart.

Start with one wire. Notice the magnetic field vectors. Pick a starting point, turn the integral on, drag the pencil around the wire counterclockwise, and come back to the starting point. What is the value of the path integral? Push the "set integral \(= 0\)" button to zero out the integral. Turn off the integral and pick a different starting point. After turning the integral back on, go around the wire again, taking a different path but going the same direction (counterclockwise). What is the path integral? Notice that with a different path the value of the magnetic field (\(\mathbf{B}\)) along the path and the direction of \(d\mathbf{l}\) are both different, but by the time you get back to your starting point the sum (integral) of \(\mathbf{B}\cdot d\mathbf{l}\) is the same. This is what Ampere's law says: that the integral around the path only depends on the current enclosed (times \(\mu_{0}\)).

What do you expect if you go in the other direction (clockwise vs. counterclockwise)? Try it (zero out the integral before each time). Note that \(d\mathbf{l}\) now points in the opposite direction. Therefore, the value of the integral is negative. The current flowing through this loop is negative with respect to the normal to the loop (since the normal to this loop is into the screen). This corresponds to a current flow out of the screen. This agrees with the result from the other (counterclockwise) path.

Now try with two wires. Again notice the magnetic field vectors. Pick a starting point and mark it. Drag the pencil around the red wire. Why is the integral the same as before? Zero the integral and drag the black dot in a circle around both wires. What is the integral? What does that mean about the total current enclosed in the circle you just made? How does the current in the blue wire compare with the current in the red wire? Since the integral is zero, we know that the currents must be equal and opposite.

If the path integral is zero, does that mean that the magnetic field along the path is zero? Why or why not? When the path integral is zero, the total current enclosed is zero, but it is not necessarily true that the magnetic field is zero. There must be a symmetry in the problem in order for Ampere's law to be useful in determining the magnetic field.

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

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