Figure 8.12: The magnetic field near the center of a single coil carrying a current I.
We can readily find examples where Ampere's Law is more useful, by virtue of being easier to apply, than the Biot-Savart Law. For example, as was the case for the electric field, it was deemed to be very practical to have a device which could "store" magnetic field as a capacitor "stores" an electric field. We know that what the capacitor really stores is electric charge separation, but one of the properties for which a parallel-plate capacitor is useful is its ability to produce a uniform electric field between its plates. The equivalent device for magnetic fields is the solenoid. It's simply a conducting wire wrapped into a cylindrical shape. Even though the actual shape of the wire is helical, for densely packed wrapping we can actually consider the solenoid to be a bunch of closely spaced coils. Near the center of each coil we know that the magnetic field is nearly perpendicular to the plane of the coil as shown in figure 8.12.
If we place coils on either side of the first and let them all carry the same current I in the same direction, then the magnetic field lines will link together near the center to make a field that is approximately uniform in direction and strength near the center of the set of coils. Placing many coils in close proximity to each other yields the solenoid field (see figure below).
Approximating the field as constant in direction and magnitude near the center of the solenoid allows us to use Ampere's Law to calculate its magnitude.
Choosing a flat rectangle as the Amperian loop,
we see that the contributions to the loop integral
can be broken into four parts. The area bounded by
the flat rectangle is penetrated by N loops over
a length L.
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We can use what we just derived to examine the case of another device that produces a uniform magnetic field. In this case, we take advantage of our analysis of the solenoid to ask what happens if we bend a solenoid into a circle so that the ends join. The new configuration still has approximately zero field in the regions outside the volume contained by the coils, but the field inside that volume is again approximately uniform if the distance between the coils is small compared to the size of the coils. This device is a toroid.
We expect the magnetic field to have circular symmetry about the center of the toroid using the same reasoning as for the solenoid. Hence, we expect it to be most useful to use circular paths for evaluating the Ampere integral of magnetic field and path. For any circular path whose area is not intersected by the coils, the magnetic field is zero and the current penetrating the area is, by definition, zero. Hence, Br < Rinner = 0. For a circular path whose radius is greater than Router, we also expect to have no magnetic field. This might seem to be a contradiction with Ampere's Law because the current in the toroid coils definitely penetrate the area contained, however, note that the coils take current through the area (out of the page) at Rinner and back into the page at Router. So, the net current penetrating the area contained by the circular loop is zero since every loop of the wire carries current through the area twice - in opposite directions! So it is consistent that the magnetic field be zero.
We can consider an instance of a current distribution which does not have cylindrical symmetry, but which is susceptible to Ampere's Law for finding the magnitude of the magnetic field. Consider a sheet of current which is infinitesimally thin but infinitely long and wide. The sheet has a linear current density (i.e. current per unit length), l.
We can replace the sheet with an infinite set of wires arranged so that each wire carries current consistent with an overall current per unit length of l as in the case of the current sheet. We see in figure 8.19 that wires which are equidistant from the line from the set of wires to point P have magnetic fields whose vertical components cancel and whose horizontal components add. Hence, the net magnetic field is in the horizontal direction. This field is uniform since the distribution of wires is infinite, i.e. any position for point P can be considered as the "middle" of an infinitely long current sheet. Note also that the magnetic field direction for the infinite sheet is independent of the distance of P from the sheet and that the same arguments for extending consideration of a finite set of wires to an infinite current sheet state that the magnetic field direction on the other side of the sheet has the field pointing in the opposite direction. Now we can apply Ampere's Law for the path shown in figure . Now we can apply Ampere's Law for the path shown in the figure below.
The loop gives us
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Send comments to larryg@upenn5.hep.upenn.edu.
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