Displacement Current

The current that exists at any point in space where a time-varying electric field (E) exists, i.e., \[\frac {dE}{dt}\] ≠ 0, is called displacement current (iₐ).

\[\frac {dE}{dt}\] ≠ 0 ⇒ i_d​ exists

\[\oint_c\vec{B}\cdot d\vec{l}=\mu_0I_c+\varepsilon_0\mu_0\frac{d\Phi_E}{dt}\]

This equation states that not only the current but also the changing electric field produces a changing magnetic field. This equation is known as the Ampere-Maxwell Circuital Law.

The Charging Capacitor Scenario

Imagine a simple circuit: a battery connected to a parallel plate capacitor through a resistor. Current ici_cic​ flows through the wire as the capacitor charges.

Ampere's Law Fails:

Apply the original Ampere's Law: \[\oint \vec B\cdot d\vec l\] = μ₀I_enc

• Surface S1 (cuts the wire): I_enc = i_c → gives a non-zero magnetic field
• Surface S2 (passes through the gap): I_enc = 0 (no charge flows through gap) → gives zero magnetic field

Think of a relay race. When the first runner (conduction current in the wire) stops, a second runner (displacement current in the gap) picks up the baton. The "race" (current loop) never breaks — it just changes form.

Between the capacitor plates, there is no net charge movement, but the electric field is changing. Maxwell proposed that a time-varying electric field must produce a magnetic effect, just as a time-varying magnetic field produces an electric field (Faraday's Law). He called this equivalent effect Displacement Current.

Step 1: Between the plates of a charging capacitor, charge qqq on each plate creates an electric field:

• \[E=\frac{q}{\varepsilon_0A}\Rightarrow\Phi_E=EA=\frac{q}{\varepsilon_0}\]

Step 2: Differentiate with respect to time:

• \[\frac{d\Phi_E}{dt}=\frac{1}{\varepsilon_0}\frac{dq}{dt}=\frac{i_c}{\varepsilon_0}\]

Step 3: Therefore:

• \[\varepsilon_0\frac{d\Phi_E}{dt}=i_c\]

This shows: Displacement current = Conduction current (equal in magnitude, maintaining current continuity).

Step 4: Maxwell modified Ampere's Law to include both:

• Total current = Conduction current + Displacement current
  i_total​ = i_c​ + i_d​
• Generalised Ampere–Maxwell Circuital Law
  \[{\oint\vec{B}\cdot d\vec{l}=\mu_0\left(i_c+\varepsilon_0\frac{d\Phi_E}{dt}\right)}\]

This corrected law now works for both surfaces S1 and S2, eliminating the inconsistency.