Physics Friday 129

Part 9: Maxwell’s Laws

Summing up so far, electrostatics uses Gauss’ law, which in differential form states
.
For magnetostatics, we had (Gauss’ law of magnetism, equivalent to saying that magnetic monopoles do not exist), and Ampère’s Law, which in differential form states
.
Lastly, we began electrodynamics with Faraday’s Law, which in differential form is

(we have used SI units for all of these).
However, these four equations, as written here, are inconsistent; one cannot expect the static equations to hold for dynamic situations. In recognizing this, and providing the correction, is where James Clerk Maxwell made his great achievement.
Specifically; the problem is in Ampère’s law. Taking the divergence of both sides,
.
but the divergence of a curl is always zero, so the left-hand side must be zero, and Ampère’s Law as formulated for magnetostatics requires . But by the continuity equation, ; so this condition only holds when the charge density is fixed.

According to J.D. Jackson in his Classical Electrodynamics textbook (my textbook for E&M at Caltech), Maxwell’s repair can be reasoned as follows: One begins with the continuity equation . One then notes that Gauss’ law says , and so, taking its time derivative, one sees that
; plugging this into the continuity equation, the result is

,
so the quantity must always be solenoidal (divergence-free); thus, one replaces J in the magnetostatic form of Ampère’s law with this quantity; then both sides are always divergence free:
.
Maxwell dubbed this added term the displacement current. While it has units of current density, it is not an actual current of flowing charges, but instead indicates that, just as a current generates a magnetic field, so does a time-varying electric field.
The four equations,
.



are collectively known as Maxwell’s equations, and are the collective basis of all classical electrodynamics.
From the last two, we see that a time-varying electric field generates a magnetic field (Ampère’s Law), and a time-varying magnetic field generates an electric field (Faraday’s Law); the combination of these two makes possible electromagnetic waves.

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