Ch 08 - Electromagnetic Waves12th Physics

Between the plates of a charging capacitor, the accumulated charge $Q$ changes over time. Consequently, the Electric Field ($E$) and Electric Flux ($\Phi_E$) across the gap are continuously changing with time.

Maxwell postulated a profound symmetry in nature: just as a changing magnetic field produces an electric field (Faraday's Law), a time-varying electric field produces a magnetic field.

This equivalent, non-physical current produced by the changing electric flux in a vacuum or dielectric is defined as Displacement Current ($I_d$). It behaves exactly like a physical conduction current in its ability to produce a magnetic field.

Ampere's Law states that the magnetic field line integral around a closed loop is strictly proportional to the conduction current ($I_c$) passing through the enclosed surface.

Maxwell identified a fatal logical flaw when considering a charging capacitor.

1. If a flat surface is drawn intersecting the wire outside the capacitor, conduction current $I_c$ passes through it, correctly predicting a magnetic field.
2. However, if a 'pot-shaped' bulging surface is drawn sharing the same boundary loop but passing between the capacitor plates (where there are no flowing charges), $I_c = 0$.

This created a logical contradiction: Ampere's law predicted a magnetic field for one surface, but exactly zero magnetic field for another surface sharing the identical boundary loop, violating the physical continuity of the magnetic field.

Variables & Units:

• I_d = Displacement current (A)
• \epsilon_0 = Permittivity of free space ($8.854 \times 10^{-12} \text{ F/m}$)
• \Phi_E = Electric flux ($\text{V}\cdot\text{m}$ or $\text{N}\cdot\text{m}^2/\text{C}$)
• t = Time (s)

Conditions/Notes:

• Exists only when the electric field (and thus flux) is changing with time.
• For a static electric field, $\frac{d\Phi_E}{dt} = 0$, hence $I_d = 0$.