General Form of Ampere’s Law
We have seen that Ampere’s law
G
G
v∫ B ⋅ ds = µ I
o enclosed
allows us to calculate B in those
C
cases of sufficient symmetry, in analogy with Gauss’s law allowing us to find E in high
symmetry situations. We mentioned that in the form above Ampere’s law is limited to constant
currents, or equivalently, to constant electric fields in the wires producing those constant
currents. About 140 years ago, Maxwell generalized this to allow for time-varying currents, or
electric fields.
S2
C
S1
To see the modification to Ampere’s law in the form above when there are time-varying
electric fields, consider an electric circuit with a capacitor in it. If we analyze Ampere’s law, in
the form above, using a closed loop around the wire with current I flowing into the capacitor (as
shown), we would set the contour integral equal to µoI. However, the enclosed current does not
continue between the capacitor plates – there is no “conduction current” in the gap between these
plates. The enclosed current is really defined as the integral of the current density over a surface
G G
S:
I enclosed = ∫∫ J ⋅ dA ,
S
or the flux of the current density. The surface S is any surface bounded by the curve C used in
the contour integral. Here’s the dilemma. If we use as a surface S1, the circular plane area that is
bounded by C, then the current enclosed will just be I, the current in the wire. If instead we use
the surface S2, also bounded by C, but going between the capacitor plates, then the conduction
current crossing this surface is zero – since there is no conduction current in the gap between the
capacitor plates – and therefore the Ienclosed is zero and we have a contradiction. Maxwell
recognized this contradiction and introduced an extra current – called a displacement current –
Id, defined as
dΦE
Id = εo
,
dt
where ΦE is the electric field flux (displacement here has nothing to do with a vector distance).
As the capacitor is being charged or discharged and its Q is changing, there will be a changing E
field within the gap between the capacitor plates and therefore there will be a displacement
current there. If we generalize Ampere’s law to include not only conduction currents, but also
displacement currents, then the above contradiction disappears. So, we have the generalization
of Ampere’s law to be
G G
dΦ
∫vC B ⋅ ds = µo ( I + I d ) = µo I + µoε o dt E .
For the case of the capacitor let’s evaluate Id. Since E = σ/εo between the capacitor plates, We
can compute ΦE = EA = σA/εo = Q/εo, where Q is the charge on the capacitor. Then Id = dQ/dt =
I, and we get the same result for the line integral of B whether we use surface S1 or S2.
This added term, involving the time rate of change of E producing B, is of
fundamental importance. At the end of the course, we will see a few of its consequences.