Step 1 (Why the idea was needed): Consider charging a capacitor. Current flows in the connecting wires, but between the plates no charge crosses. If we apply Ampere's circuital law to two surfaces bounded by the same loop, one pierced by the wire and one passing through the gap, we get contradictory results. To remove this inconsistency Maxwell introduced a second kind of current in the gap.
Step 2 (Definition): The displacement current is the current-like quantity associated with a time-varying electric field between the plates. Its size equals the rate of change of electric flux times \(\varepsilon_0\):
\[ I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \]
Step 3 (Conduction current): The familiar current in wires is the conduction current, arising from drifting free electrons. Its magnitude is the charge passing per second:
\[ I_c = \frac{dq}{dt} \]
Step 4 (Contrast in a table of ideas):
• Origin: \(I_c\) from moving charges, \(I_d\) from a changing \(\vec{E}\).
• Medium: \(I_c\) flows through conductors, \(I_d\) appears even in vacuum or a dielectric gap.
• Continuity: across a charging capacitor \(I_d\) in the gap exactly equals \(I_c\) in the wire, so the total current is continuous.
Step 5 (Unified law): Maxwell's corrected Ampere law reads \(\oint \vec{B}\cdot d\vec{l} = \mu_0 (I_c + I_d)\), showing both currents create magnetic fields on equal footing.
\[\boxed{I_d = \varepsilon_0 \dfrac{d\Phi_E}{dt}, \quad I_c = \dfrac{dq}{dt}}\]