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Illustration 28.4: Path Integral
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Ampere's law is ∫ B · dl = μoI, where the integration is over a closed loop (closed path), dl is an element of the path in the direction of the path, μo is the permeability of free space (4π x 10-7 T·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 T·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 (B) along the path and the direction of dl are both different, but by the time you get back to your starting point the sum (integral) of B · dl is the same. This is what Ampere's law says: that the integral around the path only depends on the current enclosed (times μ0).
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 dl 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.