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Displacement Current

Charge a capacitor through a resistor and a current flows in the wires, but it stops dead at the plates: no charge crosses the gap. Yet a compass placed near the gap still deflects, so a magnetic field circulates there as if a current were flowing. Maxwell's resolution was to add a term to Ampere's law: a changing electric flux acts as a current, the displacement current $I_\text{disp} = \varepsilon_0\, d\Phi_E/dt$. Because the field between the plates is $E = Q/(\varepsilon_0 A)$, this works out to $I_\text{disp} = dQ/dt = I_\text{cond}$ exactly, so the total current is continuous and an Amperian loop gives the same $B$ whether it threads the wire or the gap. That missing term is what makes light: it closes the loop between changing $E$ and $B$ that becomes an electromagnetic wave.

Figure 1. The displacement current. Top: a capacitor charges through a resistor; conduction current flows in the wires, the electric field builds in the gap, and a slidable Amperian loop encloses the conduction current at the wire or the displacement current at the gap, the same value. Bottom: the conduction and displacement currents versus time, which coincide exactly, while the gap field rises. Method: closed-form RC charging with I_disp = eps0 dPhi_E/dt. Source: Griffiths, Introduction to Electrodynamics, 5th ed., Sec. 7.3.
resistance R2.0
capacitance C1.5
loop: on wire18 %

WHAT TO TRY

  • Slide the Amperian loop from a wire into the gap: the enclosed current $I_\text{enc}$ stays the same, because the displacement current in the gap exactly replaces the conduction current in the wire.
  • Watch the bottom plot: the conduction current (cyan) and the displacement current (dashed green) lie on top of each other at every instant, both decaying as $e^{-t/RC}$.
  • Raise the capacitance or the resistance: the time constant $RC$ grows, the charging slows, and both currents decay more gently while the gap field rises more slowly.
  • Note that the current is largest at the start (the gap field is changing fastest) and falls to zero once the capacitor is charged and the field is steady.