Week 7: Sources of the Magnetic Field 249
It is now simple to apply Ampere’s Law, as we get no contribution from the sides ofCand equal
positive contributions from the upper and lower legs ofC:
∮
CB~·d~ℓ = μ 0 Ithru C2 B||b = μ 0 λbB|| =μ 0 λ
2(543)
whereB||is the magnitude of the component ofB~ parallel to the sheet a distancey/2 above or
below it. Of course we note that this fielddoesn’t depend onyso the field above and below the sheet
isuniformto the right and left respectively.
There is a bit of insight to be gained from thinking abouttwosheets, one carrying current in,
one carrying current out, separated by a distanced. In this case the superposition principle suggests
that the field above the two sheets and below the two sheets will be zero, as the contributions from
the two sheets cancel. In between, though, theyaddto a total magnitude of:
B||=μ 0 λ (544)If we imagine thatλis made up of the field in a lot of very closely spaced single wires each carrying
some currentI, then you can see that:
λ=nI (545)
or, the number ofwiresper unit length times the current per wire equals the amount ofcurrentper
unit length. The field in between is thus:
B||=μ 0 λ=μ 0 nI (546)which looksjust like the field of a solenoid!
Recall that our computation of the field inside an infinitely long solenoiddidn’t depend on the
cross-sectional shape of the solenoid. In fact, it could have beenrectangular! If we imagine that the
top and bottom sides of the rectangle get longer and longer, eventually we can imagine that they
becomeinfinitelylong and close onlyat±∞so that the current that goes in at the top returns on
the bottom (say). In this way we can see that our result for the pair of infinite sheetsmakes sense
and is completely consistent. We could have guessed this result by mentally deforming a solenoid
until it looked in our minds like two infinite sheets in close to where we were actually measuring the
field.
It also tells us that even though we have been quite careful to makethe sheets we have been
considering be planar, all we really need is for them to bestraightin the left-right direction, and
continue on to infinity (and “close”) there in the direction in and out of the page. Two e.g. hyperbolic
sheets of current that stretch to infinity would have exactly the same field in between them as we
obtained in this example. This sort of conceptual understanding can be very useful later on, as can
the ability to think in terms oftopological deformationsof the sort we have just considered, so don’t
be surprised if a quiz question probes whether or not you “get it” well enough to answer simple
conceptual questions.
7.7: Summary
Yes, this week is long enough, and has enough content, that it is worth a summary. We have covered
one and a half Maxwell equations, after all!
At this point you should be aware that unless and until somebody positively discovers magnetic
monopoles in an experimentally reproducible setting so that everybody agrees that they are real