248 Week 7: Sources of the Magnetic Field
current through the any surface bounded byCis zero, as every wire goes (at best) into the surface
one time and right back out out one time.
Our conclusion is that the toroidal solenoidconfinesthe magnetic field to live inside the torus,
and the geometry of the field causes it to drop off like 1/r! How useful! How interesting! Solenoids
in general seem to like totrapmagnetic field lines and keep them from escaping. If we bend them
around in curves, they keep the field inside (and cause it to vary by getting weaker on the outside
edges of the curves). If we wrap them back into themselves (making a torus or a topological knot of
some sort then the magnetic field cannot get out into the room and remains confined to the inside
of the coil.
This property will turn out to be very useful next week when we consider making inductors
out of solenoids, as a toroidal solenoid will have the helpful property of havingvery littlemutual
inductance with nearby current loops, where finite length regular solenoids produce a pesky “fringe
field” at their ends that can induce unwanted voltages in conductors or loops close to those ends.
If you look inside a computer or other electronic device, you will usually see a few toroidal
inductors soldered into the motherboard, and that is exactlywhythey are shaped the way they are
shaped – it is very “bad” for computer motherboards to pick up inductive signals from processes that
have nothing to do with their function, especially if the voltages involved approach the threshold
that can trigger flips and flops in its enormously complex bit processing structure.
Example 7.6.5: Infinite Sheet of Current
y/2y/2λinbBBCFigure 90: A side view of an infinite sheet of conductor carrying a current (per unit length)λinto
the page. The field due to the sheet is symmetric up and below the sheet as drawn, and must point
parallel to the sheet because every point is in the middle of the infiniteplane (as usual). Any up-
down asymmetry would violate mirror symmetry about that “middle” because the problem would
not change but the solution would. This leads us to the Amperian Pathshown, which should remind
you of that of the infinte solenoid, with sides perpendicular to the field.
In figure 90 we see our final example, an infinite conducting sheet ofnegligible thickness (exag-
gerated in the picture) carrying a uniformcurrent per unit transverse lengthinto the paper. We then
follow a familiar ritual. Every point is in the middle of an infinite sheet, so our picture is located in
the middle. If we flip the picture over (maintaining the direction of thecurrent into the paper) the
field has to be the same, so we know that the field has to have the samemagnitudeequal distances
above and below the plane. We know that the picture hasmirror symmetry around any vertical
line. We know that there is much current to the right of that line (which produces a field with an
upward directed component above the sheet) as there is to the left of the line (which produces a field
with a symmetric downward directed component), so our right handtells us that the only possible
direction for the field is to the rightparallelto the sheet above it, and to the left parallel to the
sheet below it. A sensible Amperian Path is then a rectangle symmetricabout the sheet with sides
perpendicular to the field and ends parallel to it, traversed in the right handed direction as shown.