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Displacement Current and Ampère’s Law

Displacement Current and Ampère’s Law.
One more addition needs to be made to the governing equations of electromagnetics before we are finished. Specifically, we need to clean up a glaring inconsistency. From Ampère’s law in magnetostatics, we learned that
Taking the divergence of this equation gives

However, as is shown on p. 188 in Ch. 5 the continuity equation (conservation of charge) requires that

We can see that (2) and (3) agree only when there is no time variation or no free charge density. [This makes sense since (2) was derived only for magnetostatic fields in Ch. 7.] Ampère’s law in (1) is only valid for static fields and, consequently, it violates the conservation of charge if we try to directly use it for time varying fields.
Ampère’s Law for Dynamic Fields
Well, what is the correct form of Ampère’s law for dynamic (time varying) fields? Enter James Clerk Maxwell (ca. 1865) – The Father of Classical Electromagnetism. He combined the results of Coulomb’s, Ampère’s, and Faraday’s laws and added a new term to Ampère’s law to form the set of fundamental equations of classical EM called Maxwell’s equations. It is this addition to Ampère’s law that brings it into congruence with the conservation of charge law. Maxwell’s contribution was to modify Ampère’s law (1) to read

The second term on the right-hand side is called the displacement current density. We will investigate this in more detail shortly.
Consistency with Conservation of Charge
First, however, let’s see that adding this term in (4) makes it consistent with conservation of charge. First, we take the divergence of (4)

Next, from Gauss’ law
Ñ `D=p
we take the time derivative yielding

Substituting (6) into (5) gives

This is just the continuity equation (3). Amazing! With this additional displacement current term in (4), Ampère’s law is now consistent with conservation of charge. Of course, this is not a proof that (4) is now the correct form of Ampère’s law for dynamic fields, although it is. The correctness of (4) is essentially an experimentally derived proof as with
.Gauss’ law,
.Ampère’s force law, and
. Faraday’s law earlier in this chapter.
Integral Form of Ampère’s Law
The integral form of Ampère’s law is obtained by integrating (4) over some arbitrary open surface s

and applying Stokes’ theorem to give

The first term on the right hand side is free current. The second term is this new displacement current.
What is Displacement Current?
So what is displacement current? Consider an example of a simple electrical circuit containing a sinusoidal voltage source and a capacitor:

From our experience in the lab, at DC the current I = 0. However, if f ¹ 0, we will observe and measure some nonzero time varying current I. However, if free charge (i.e., electrons) cannot “jump” the gap between the capacitor plates, then how can there be current in the circuit? Because there is another type of current that completes the circuit. Between the plates of the capacitor exists the displacement current. Consider just the capacitor:

You learned previously in EE 381 that under static conditions
Q = CV
If the voltage source is assumed to vary “slowly enough” with time, then (9) is still valid
QT » CV(t)
In words, (10) reveals to us that as V changes with time, so does the stored charge on the conducting plates of the capacitor. Now, V (t)as we’ve seen is not uniquely defined. Nevertheless, if the distance between the plates is sufficiently small (so that E(t) is essentially quasi-static) it may be a very good approximation. Then

Taking the time derivative of (10) gives

We interpret this equation as equating two types of current

That is, (11) can be written as

which should be an extremely familiar equation This displacement current has “completed” the electrical circuit

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