Related Concepts: 4.01 Charge Conservation & Continuity Equation Mechanics | 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem | 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation

4.02 Faraday’s Law of Induction & Maxwell’s Displacement Current

Core Idea

Two symmetric statements turn static field theory into electrodynamics:

• Faraday (experiment, 1831): a time-varying magnetic field creates a circulating electric field.
• Maxwell (pure mathematics, 1865): a time-varying electric field must create a circulating magnetic field — otherwise Ampère’s law contradicts conservation of charge.

Faraday’s half was discovered in a laboratory. Maxwell’s half was discovered on paper, purely to repair a mathematical inconsistency — and it predicted electromagnetic waves before anyone had ever produced one.

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1. Faraday’s Law of Electromagnetic Induction

1.1 Statement and Equation

Faraday's Law

The induced electromotive force (EMF) around any closed circuit equals the negative time rate of change of the total magnetic flux $\Phi$ linking that circuit.

$e={\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d\Phi }{dt}=-\frac{d}{dt}{\int }_S\vec{B}\cdot d\vec{s}$

Terminology & Concept Breakdown

• Electromotive force ($e$): despite the name, not a force — it is the work done per unit charge in carrying that charge once around the closed loop $C$. Units: volts.
• Magnetic flux ($\Phi$): the total amount of $\vec{B}$ threading through the open surface $S$ that the loop $C$ bounds. Units: webers (Wb).
• Lenz’s Law (the minus sign): the induced current flows in whatever direction produces a magnetic flux that opposes the change that caused it. Nature resists change in flux. Without this minus sign you would have a machine that amplifies its own output — a violation of energy conservation.

The single most important structural consequence

In electrostatics, ${\oint }_C\vec{E}\cdot d\vec{l}=0$ — the electric field was conservative. Faraday’s law says that once $\vec{B}$ changes with time, this line integral is no longer zero. The dynamic electric field is non-conservative: it circulates. This is precisely why $\nabla \times \vec{E}=0$ (static) must be upgraded to $\nabla \times \vec{E}=-\partial \vec{B}/\partial t$ (dynamic).

1.2 Differential (Point) Form

Apply Stokes’s Theorem to the left-hand side and assume a stationary loop so the derivative may be taken inside as a partial derivative:

${\int }_S(\nabla \times \vec{E})\cdot d\vec{s}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$

Since this holds for any arbitrary surface $S$, the integrands must be equal:

Faraday's Law — Differential Form (Maxwell's 3rd Equation)

$\boxed{\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}}$

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2. The Three Operational Cases of Induced EMF

Flux $\Phi ={\int }_S\vec{B}\cdot d\vec{s}$ can change for two independent reasons: $\vec{B}$ itself changes, or the loop moves/deforms. This gives three cases.

[FIGURE: Three side-by-side panels — (a) stationary loop with a time-varying B field through it (transformer EMF); (b) rigid loop sliding with velocity u through a uniform static B field (motional EMF); (c) moving loop in a time-varying B field (combined). — source: Cheng 2nd Ed., Figs. 7-2 to 7-4 / lecture slide L 9.pdf]

Case 1 — Transformer EMF (stationary loop, time-varying $\vec{B}$)

$e_{trans}=-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}⟺\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$

The loop sits still; only the field changes. Physical example: an ordinary transformer — the secondary winding never moves, but the alternating primary current makes $\vec{B}$ in the core oscillate.

Case 2 — Motional EMF (moving loop, static $\vec{B}$)

$e_{motion}={\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}$

The field is frozen in time; the conductor cuts through it at velocity $\vec{u}$. The force on the carriers comes straight from the magnetic part of the Lorentz force, $\vec{F}=q(\vec{u}\times \vec{B})$. Physical example: a DC generator, or a conducting bar sliding along rails.

Case 3 — Combined EMF (moving loop, time-varying $\vec{B}$)

$e_{total}=\underset{\text{transformer part}}{\underbrace{-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}}}+\underset{\text{motional part}}{\underbrace{{\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}}}$

The most general case — both mechanisms contribute and simply add.

Case	Loop	$\vec{B}$ field	EMF expression	Everyday device
Transformer	Stationary	Time-varying	$-{\int }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$	Transformer, induction cooktop
Motional	Moving ($\vec{u}$)	Static	${\oint }_C(\vec{u}\times \vec{B})\cdot d\vec{l}$	DC generator, dynamo
Combined	Moving ($\vec{u}$)	Time-varying	Sum of both	AC machine with a rotating rotor

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3. Master Derivation: Maxwell’s Displacement Current

[PYQ: 2016, 2017] for the derivation; the definition alone also appeared [PYQ: 2023] as a short-note term.

PYQ tagging note

Displacement current is asked as a standalone question in 2016 and 2017 only (“Define displacement current. Determine the displacement current in between two parallel plates of a capacitor energized by an alternating current source.”). It also appears as a required step inside the “Deduce Maxwell’s equations” question [PYQ: 2018, 2019, 2023] — see 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem. So its real exam weight is higher than the two standalone years suggest.

3.1 The Fatal Flaw in Ampère’s Static Law

In magnetostatics, Ampère’s circuital law in differential form is:

$\nabla \times \vec{H}=\vec{J}$

Take the divergence of both sides:

$\nabla \cdot (\nabla \times \vec{H})=\nabla \cdot \vec{J}$

The left side is identically zero by the null identity (the divergence of the curl of any vector field is always zero, $\nabla \cdot \nabla \times \vec{A}\equiv 0$). Therefore Ampère’s static law forces:

$0=\nabla \cdot \vec{J}$

But from 4.01 Charge Conservation & Continuity Equation Mechanics, the continuity equation says:

$\nabla \cdot \vec{J}=-\frac{\partial {\rho }_v}{\partial t}\ne 0\text{whenever charge density changes with time}$

The contradiction

Ampère’s static law demands $\nabla \cdot \vec{J}=0$ everywhere and always. Conservation of charge says $\nabla \cdot \vec{J}\ne 0$ whenever ${\rho }_v$ varies. Both cannot be true. Since conservation of charge is an experimental law of nature that cannot be abandoned, Ampère’s law must be the thing that is incomplete.

3.2 Maxwell’s Repair

Maxwell postulated a missing term — an unknown displacement current density ${\vec{J}}_d$ — added to the right-hand side:

$\nabla \times \vec{H}=\vec{J}+{\vec{J}}_d$

Step 1. Take the divergence of both sides. The left side is still identically zero:

$0=\nabla \cdot \vec{J}+\nabla \cdot {\vec{J}}_d⟹\nabla \cdot {\vec{J}}_d=-\nabla \cdot \vec{J}$

Step 2. Substitute the continuity equation $\nabla \cdot \vec{J}=-\partial {\rho }_v/\partial t$:

$\nabla \cdot {\vec{J}}_d=\frac{\partial {\rho }_v}{\partial t}$

Step 3. Substitute Gauss’s postulate ${\rho }_v=\nabla \cdot \vec{D}$:

$\nabla \cdot {\vec{J}}_d=\frac{\partial }{\partial t}\left(\nabla \cdot \vec{D}\right)$

Step 4. Spatial and temporal derivatives are independent, so their order may be exchanged:

$\nabla \cdot {\vec{J}}_d=\nabla \cdot \left(\frac{\partial \vec{D}}{\partial t}\right)$

Step 5. Strip the divergence operator from both sides:

Key Result — Displacement Current Density

$\boxed{{\vec{J}}_d=\frac{\partial \vec{D}}{\partial t}=ϵ\frac{\partial \vec{E}}{\partial t}}[{\text{A/m}}^2]$

and Ampère’s law becomes the Ampère–Maxwell Law: $\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$

Definition to memorise

Displacement current is a fictitious current, proportional to the time rate of change of the electric flux density $\vec{D}$, introduced by Maxwell to make Ampère’s circuital law consistent with the conservation of charge in time-varying fields. It involves no motion of physical charge, yet it produces a magnetic field exactly as a real conduction current does.

graph TD
    A["Ampère's static law<br/>∇×H = J"] -->|"take divergence"| B["∇·J = 0<br/>(forced)"]
    C["Conservation of charge<br/>∇·J = −∂ρᵥ/∂t ≠ 0"] --> D{"CONTRADICTION"}
    B --> D
    D -->|"Maxwell adds Jd"| E["∇×H = J + Jd"]
    E -->|"divergence + continuity<br/>+ Gauss ρᵥ = ∇·D"| F["Jd = ∂D/∂t<br/>DISPLACEMENT CURRENT"]
    F --> G["∇×H = J + ∂D/∂t<br/>AMPÈRE–MAXWELL LAW"]
    G --> H["Time-varying E creates H<br/>→ EM waves possible"]

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4. High-Yield Derivation: Displacement Current in a Capacitor

[PYQ: 2016, 2017 — 07/09 Marks] — this is the standard second half of the displacement-current question.

The physical puzzle: current flows in the wires feeding a capacitor, yet the gap between the plates is an insulator where no charge can cross. If you draw an Ampèrean loop around the wire and stretch its surface so it passes between the plates instead of cutting the wire, the enclosed conduction current is zero — so where did the magnetic field go? Displacement current is the answer.

[FIGURE: Parallel-plate capacitor of plate area A and separation d energised by an AC source v(t) = V₀ sin ωt. Show conduction current i_c in the connecting wires, E-field lines between the plates, and two alternative Ampèrean surfaces bounded by the same loop — one cutting the wire, one passing through the gap. — source: Cheng 2nd Ed., Fig. 7-1 / "01 Static Electric Field.pdf"]

Step 1. Energise a parallel-plate capacitor (plate area $A$, separation $d$, dielectric $ϵ$) with a time-harmonic source:

$v(t)=V_0\sin \omega t$

Step 2. Neglecting fringing, the field between the plates is uniform:

$\vec{E}(t)={\hat{a}}_n\frac{v(t)}{d}={\hat{a}}_n\frac{V_0}{d}\sin \omega t$

Step 3. The electric flux density in the dielectric:

$\vec{D}(t)=ϵ\vec{E}(t)={\hat{a}}_n\frac{ϵV_0}{d}\sin \omega t$

Step 4. Differentiate to get the displacement current density:

${\vec{J}}_d=\frac{\partial \vec{D}}{\partial t}={\hat{a}}_n\frac{ϵV_0\omega }{d}\cos \omega t$

Step 5. Integrate over the plate area $A$ to get the total displacement current:

$i_d={\int }_A{\vec{J}}_d\cdot d\vec{s}=J_{d}A=\frac{ϵA}{d}{V}_0\omega \cos \omega t$

Step 6. Recognise the parallel-plate capacitance $C=\frac{ϵA}{d}$:

$i_d=C{V}_0\omega \cos \omega t=C\frac{dv(t)}{dt}$

Step 7. From elementary circuit theory, the conduction current in the external wires is:

$i_c(t)=C\frac{dv(t)}{dt}$

Conclusion

$\boxed{i_d(t)=i_c(t)}$ The displacement current between the plates is exactly equal to the conduction current in the wires. Total current is therefore continuous around the whole circuit loop, and Ampère’s law gives the same magnetic field regardless of which surface you stretch across the loop.

The sentence that earns the last mark

“Displacement current is not a flow of physical charge — it is a time-varying electric field that behaves, magnetically, exactly as a real current does. It is what bridges the insulating gap and keeps total current continuous.”

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5. Comparison: Conduction vs. Displacement Current

Property	Conduction Current $\vec{J}=\sigma \vec{E}$	Displacement Current ${\vec{J}}_d=\partial \vec{D}/\partial t$
Physical nature	Actual drift of free charge carriers	No charge motion; changing electric flux
Requires	A conducting medium ($\sigma \ne 0$)	Any medium including vacuum
Governing relation	Ohm’s law, $\vec{J}=\sigma \vec{E}$	${\vec{J}}_d=ϵ\partial \vec{E}/\partial t$
Exists in DC/static?	Yes	No — vanishes when $\partial /\partial t=0$
Energy behaviour	Dissipates as ohmic heat $\vec{J}\cdot \vec{E}$	Stores/returns energy (reactive)
Dominates when	$\sigma \gg \omega ϵ$ (good conductor)	$\sigma \ll \omega ϵ$ (good dielectric)
Produces $\vec{H}$?	Yes	Yes — identically

Forward link

The ratio of these two currents, $\frac{\sigma }{\omega ϵ}$, is the loss tangent, and it is the single number that decides whether a medium behaves as a conductor or a dielectric at a given frequency. See 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations.

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6. Why This Chapter Produces Waves

Put the two halves together and note the feedback loop:

$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\text{(changing }\vec{B}\Rightarrow \text{ circulating }\vec{E})$ $\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}\text{(changing }\vec{E}\Rightarrow \text{ circulating }\vec{H})$

Each field regenerates the other. In a source-free region ($\vec{J}=0$) this becomes self-sustaining — the disturbance no longer needs its source and propagates away as an electromagnetic wave. That is the entire content of 5.01 Wave Equations & Helmholtz Equations in Source-Free Media.

Without displacement current, there are no radio waves

If Maxwell had not added $\partial \vec{D}/\partial t$, the feedback loop would be broken in one direction and electromagnetic waves would be mathematically impossible. Every wireless technology traces back to this one term.

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7. Common Mistakes That Cost Marks

Avoid these

1. Omitting the minus sign in Faraday’s law — it is Lenz’s law, and it is worth a mark on its own.
2. Calling displacement current a “real current”. It is explicitly fictitious — no charge moves. Say so.
3. Skipping the contradiction step. The whole point of the derivation is that $\nabla \cdot (\nabla \times \vec{H})\equiv 0$ clashes with continuity. If you jump straight to ${\vec{J}}_d=\partial \vec{D}/\partial t$ you lose most of the marks.
4. Forgetting to prove $i_d=i_c$ in the capacitor problem. The equality is the answer; the algebra before it is only setup.
5. Confusing transformer EMF with motional EMF. Transformer = loop still, field changes. Motional = field still, loop moves.

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8. PYQ Bank — Verbatim Questions & Answer Plans

Q1 — Displacement current, definition + capacitor [PYQ: 2016, 2017 — 07/09 Marks]

“Define displacement current. Determine the displacement current in between two parallel plates of a capacitor energized by an alternating current source.”

Answer plan (2 marked parts):

1. Definition — the boxed definition in Section 3.2. Mention explicitly that it is fictitious and that it exists to reconcile Ampère’s law with continuity.
2. Derivation — all seven steps of Section 4, ending with $i_d=i_c$ and the closing interpretation sentence. Bonus marks: briefly showing the $\nabla \cdot (\nabla \times \vec{H})=0$ contradiction (Section 3.1) demonstrates you know why ${\vec{J}}_d$ exists, not just what it equals.

Q2 — Displacement current density as a short-note term [PYQ: 2023 — part of 09 Marks]

“Briefly discuss the following terms: i) Intrinsic impedance, ii) Complex permittivity, iii) Displacement current density.”

Answer plan: For part (iii) only — 3–4 lines: the definition, the formula ${\vec{J}}_d=\partial \vec{D}/\partial t=ϵ\partial \vec{E}/\partial t$, units A/m², and the one-line reason it was introduced. Parts (i) and (ii) belong to 5.02 Plane Waves in Lossless vs. Lossy Media & Skin Depth Calculations and 1.04 Media Properties & Conductor-Insulator Behaviour.

Q3 — Faraday's law inside the Maxwell deduction [PYQ: 2018, 2019, 2023 — 08/10/13 Marks]

“Derive Maxwell’s equation from fundamental electrostatic and magnetostatic expressions by incorporating Faraday’s law of electromagnetic induction and continuity equation.” (2019 wording)

Answer plan: Faraday’s law (Section 1) and the displacement current derivation (Section 3) are both required as sub-steps. The full four-equation deduction is set out in 4.03 Maxwell’s Equations (Differential & Integral Forms) & Poynting Theorem §3.

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9. Self-Check Before Moving On

• Can you state Faraday’s law in both integral and differential form, and explain the minus sign?
• Can you name and write the EMF expression for all three induction cases, with a device example for each?
• Can you show, in four lines, exactly why Ampère’s static law contradicts conservation of charge?
• Can you derive ${\vec{J}}_d=\partial \vec{D}/\partial t$ from that contradiction without looking?
• Can you prove $i_d=i_c$ for an AC-driven capacitor?
• Can you explain in one sentence why electromagnetic waves would not exist without the ${\vec{J}}_d$ term?

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Source: 04 time_varying_fields_and_maxwells_equations.md (master dump), ECE 2105 Syllabus Week 9, PYQ bank 2015–2025, L 9.pdf, 01 Static Electric Field.pdf