🚀 Why Talk About Displacement Current?

An electric current makes a magnetic field. 💡 Maxwell noticed that for everything to stay consistent, a changing electric field must also create a magnetic field. This idea explains radio waves, visible light, and the whole electromagnetic spectrum. :contentReference[oaicite:0]{index=0}

🔋 The Charging-Capacitor Thought Experiment

Imagine a capacitor charging in a circuit. Outside the plates a conduction current i(t) flows in the wire, so Ampere’s circuit law says \( \displaystyle\oint\!\mathbf{B}\!\cdot\!d\mathbf{l}= \mu_{0}\, i(t) \)  (Eq. 8.1). :contentReference[oaicite:1]{index=1}

Pick a circular loop around the wire (radius r). Symmetry gives \( B\, (2\pi r)=\mu_{0}\,i(t) \)  (Eq. 8.2). :contentReference[oaicite:2]{index=2}

Now stretch a “pot-shaped” surface that slips between the plates—no charge crosses it, so the conduction current term is zero, yet our loop hasn’t changed. Uh-oh! Contradiction. Something is missing. :contentReference[oaicite:3]{index=3}

✨ Enter the Displacement Current

Between the plates sits only the electric field E. The electric flux through the surface is \( \Phi_E =\dfrac{Q}{\varepsilon_{0}} \)  (Eq. 8.3). If the charge Q changes, \( \displaystyle \varepsilon_{0}\,\frac{d\Phi_E}{dt}=i \)  (Eq. 8.4). That new term behaves like a current even though no charges move across the gap. We call it the displacement current: \( i_d = \varepsilon_{0}\,\dfrac{d\Phi_E}{dt}. \) 🎉 :contentReference[oaicite:4]{index=4}

🧲 The Ampere–Maxwell Law

Maxwell summed the ordinary (conduction) current ic and the displacement current id: \( i = i_c + i_d \)  (Eq. 8.5). :contentReference[oaicite:5]{index=5}

The updated circuit law becomes \( \displaystyle\oint\!\mathbf{B}\!\cdot\!d\mathbf{l}= \mu_{0}\Bigl(i_c + \varepsilon_{0}\,\frac{d\Phi_E}{dt}\Bigr) \)  (Eq. 8.6). Now every surface bounded by the same loop gives the same magnetic field—problem solved! :contentReference[oaicite:6]{index=6}

⚡ Conduction vs Displacement—Who Goes Where?

  • Outside the capacitor plates: only conduction current \( (i_c=i,\; i_d=0) \).
  • Between the plates: only displacement current \( (i_d=i,\; i_c=0) \). :contentReference[oaicite:7]{index=7}
  • Both kinds create magnetic fields with equal status. In regions with no free charges (like deep space), a time-varying electric field alone can still produce a magnetic field. :contentReference[oaicite:8]{index=8}

🔄 Perfect Teamwork of E and B

Faraday showed that a changing magnetic field makes an electric field. The displacement current shows the flip side: a changing electric field makes a magnetic field. These two facts let time-varying fields regenerate each other and race through space as electromagnetic waves. 🌊 :contentReference[oaicite:9]{index=9}

📜 Maxwell’s Equations in Vacuum

  1. \( \displaystyle\oint\!\mathbf{E}\!\cdot\!d\mathbf{A}=\dfrac{Q}{\varepsilon_{0}} \) — Gauss (electric). :contentReference[oaicite:10]{index=10}
  2. \( \displaystyle\oint\!\mathbf{B}\!\cdot\!d\mathbf{A}=0 \) — Gauss (magnetic).
  3. \( \displaystyle\oint\!\mathbf{E}\!\cdot\!d\mathbf{l}=-\dfrac{d\Phi_B}{dt} \) — Faraday. :contentReference[oaicite:11]{index=11}
  4. \( \displaystyle\oint\!\mathbf{B}\!\cdot\!d\mathbf{l}= \mu_{0} i_c + \mu_{0}\varepsilon_{0}\dfrac{d\Phi_E}{dt} \) — Ampere–Maxwell. :contentReference[oaicite:12]{index=12}

🎯 High-Yield Ideas for NEET

  • Formula for displacement current \( i_d = \varepsilon_{0}\dfrac{d\Phi_E}{dt} \).
  • The complete Ampere–Maxwell law and its surface-independent nature.
  • Role of displacement current inside a charging capacitor.
  • Mutual creation of electric and magnetic fields → electromagnetic waves.
  • Compact set of four Maxwell equations in vacuum—foundation for modern electromagnetism.

Keep practicing, and let the fields be with you! ⚡🧲