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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