ELECTROMAGNETISM & RELATIVITY - Chapter 6, Exercise 2 Solution
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Conceptual: Displacement Current and Wave Symmetry

PROBLEM
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Explain, in your own words, what "displacement current" actually is,
and why adding it to Ampere's law (Chapter 3-4) was necessary for
Maxwell's four equations to have the symmetry needed to predict a
genuinely self-sustaining electromagnetic wave.

SOLUTION
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Ampere's original law (Chapter 3-4) stated that an electric CURRENT
- actual moving charge - creates a magnetic field circling around it.
This was true and well-confirmed, but it required an actual current
of real, physically moving charge to be present.

Maxwell's real, original insight was that a CHANGING electric field,
even in a region where no actual current is flowing at all (such as
the empty space between the plates of a charging capacitor), also
produces a magnetic field - behaving, for the purposes of the
magnetic effect it creates, exactly as if a real current were
flowing there. Maxwell called this effective, non-physical "current"
the displacement current.

This mattered for symmetry because, without it, the four equations
had a genuine one-way gap: Faraday's law (Chapter 5) already
established that a changing MAGNETIC field creates an ELECTRIC
field. But before Maxwell's addition, the reverse effect - a
changing ELECTRIC field creating a MAGNETIC field - was only true
when actual current happened to be present, not in general. Adding
displacement current completed the missing half of the symmetry: now
BOTH directions work purely from changing fields alone, with no
actual moving charge required at all.

This exact two-way symmetry is what makes a self-sustaining
electromagnetic wave possible: a changing electric field creates a
magnetic field (via displacement current), which itself is now
changing and so creates a new electric field (via Faraday's law),
which is itself changing and creates another magnetic field, and so
on indefinitely - each field regenerating the other as the wave
propagates outward through completely empty space, needing no
current or charge anywhere along its own path.

ANSWER: Displacement current is the real effect by which a changing
electric field produces a magnetic field, even with no actual moving
charge present. Adding it to Ampere's law completed a genuine,
two-way symmetry with Faraday's law - without it, only a changing
magnetic field could create an electric field, but not the reverse in
general, which would have made a self-sustaining wave (each field
continuously regenerating the other) mathematically impossible.

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WHY THIS WORKS AS AN ANSWER

This connects the chapter's own compare-table (each of Maxwell's four
equations tied back to an earlier chapter) directly to the deeper
reason Maxwell's own real contribution mattered: it wasn't simply
adding a fourth equation to complete a set, but specifically
restoring a missing symmetry that turned four separate, previously-
known facts about static or current-driven fields into a theory
capable of predicting a genuinely new physical phenomenon - a
self-propagating wave requiring no ongoing source at all.