- #1
FranzDiCoccio
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- 41
I was thinking of a standard, high school level discussion of the displacement current. The need for introducing this quantity is demonstrated by considering a circuit with a charging capacitor, and (for the sake of simplicity) a circular loop whose axis is along the (straight) wire carrying the current to one of the capacitor plates. There is no dielectric in the capacitor.
Using (again, for the sake of simplicity) Biot and Savart's law one shows that the circulation of the magnetic field around the loop is non zero, because the wire is carrying a current.
However, there seems to be a paradox with (standard) Ampere's Law. Indeed, the circulation of the magnetic field should be proportional to the charge current piercing any surface spanned by the loop. But this works only if one chooses a surface pierced by the wire. If, instead, one chooses surface "enclosing" the capacitor plate attached to the wire, there is no charge current.
Thus Ampere's law seems to "fail", because the circulation of the field is clearly non-zero.
This is solved by including a "displacement current" term in Ampere's law. This displacement current is proportional to the flux of the electric field through the surface. For the first surface, there is no electric field and hence no displacement current. For the second surface there is no current, but the flux of the electric field provides the displacement current.
In all of the discussions I've found, the surfaces are not entirely arbitrary, though. The "second" surface always entirely encloses the capacitor plate.
So I was wondering: "what if the second surface is still not pierced by the wire, but encloses only a portion of the plate?".
To make things more definite, think of this situation: S1 and S2 form the surface of a truncated cone. S1 is the circle enclosed by the loop, and it is also the "bottom lid" of the truncated cone. S2 is formed by the lateral surface and the top lid of the cone, which is between the capacitor plates.
This is basically described by this figure on the wikipedia page about the displacement current. In this case, S2 entirely encloses the capacitor plate.
Now think a surface S3 that is qualitatively similar to S2, but with a much smaller top lid, so that its lateral surface intersects the plate.
Since the flux through S3 is smaller than that through S2, there would be a displacement current but, unlike the previous case, it would be smaller than the charge current, and hence could not account for the entire circulation of the magnetic field by itself.
The only answer I could think of is that there should be a residual charge current flowing radially through the capacitor plate.
In other terms, the surface is not pierced by the wire, but by the plate which, in a way, also carries a charge current.
Hence in this case Ampere's law would have both terms, a current charge term and a displacement term.
Am I making any sense?
Using (again, for the sake of simplicity) Biot and Savart's law one shows that the circulation of the magnetic field around the loop is non zero, because the wire is carrying a current.
However, there seems to be a paradox with (standard) Ampere's Law. Indeed, the circulation of the magnetic field should be proportional to the charge current piercing any surface spanned by the loop. But this works only if one chooses a surface pierced by the wire. If, instead, one chooses surface "enclosing" the capacitor plate attached to the wire, there is no charge current.
Thus Ampere's law seems to "fail", because the circulation of the field is clearly non-zero.
This is solved by including a "displacement current" term in Ampere's law. This displacement current is proportional to the flux of the electric field through the surface. For the first surface, there is no electric field and hence no displacement current. For the second surface there is no current, but the flux of the electric field provides the displacement current.
In all of the discussions I've found, the surfaces are not entirely arbitrary, though. The "second" surface always entirely encloses the capacitor plate.
So I was wondering: "what if the second surface is still not pierced by the wire, but encloses only a portion of the plate?".
To make things more definite, think of this situation: S1 and S2 form the surface of a truncated cone. S1 is the circle enclosed by the loop, and it is also the "bottom lid" of the truncated cone. S2 is formed by the lateral surface and the top lid of the cone, which is between the capacitor plates.
This is basically described by this figure on the wikipedia page about the displacement current. In this case, S2 entirely encloses the capacitor plate.
Now think a surface S3 that is qualitatively similar to S2, but with a much smaller top lid, so that its lateral surface intersects the plate.
Since the flux through S3 is smaller than that through S2, there would be a displacement current but, unlike the previous case, it would be smaller than the charge current, and hence could not account for the entire circulation of the magnetic field by itself.
The only answer I could think of is that there should be a residual charge current flowing radially through the capacitor plate.
In other terms, the surface is not pierced by the wire, but by the plate which, in a way, also carries a charge current.
Hence in this case Ampere's law would have both terms, a current charge term and a displacement term.
Am I making any sense?