00 Chapter 4 Active-Recall Diagnostic Quiz (Time-Varying Fields & Maxwell’s Equations)
> [!abstract] Overview
> Test your conceptual understanding and mathematical recall of charge conservation, continuity equation, Faraday’s law of induction, displacement current density J_d, complete Maxwell’s equations, and Poynting’s power flow theorem.
Question 1: Continuity Equation Differential Form
State the differential form of the continuity equation for electric current.
> [!faq]- Solution
> ∇ · J = -∂ρ_v / ∂t
> The divergence of current density equals the negative rate of change of volume charge density.
Question 2: Steady Current Case
What does the continuity equation reduce to for steady (direct) currents (∂ρ_v / ∂t = 0), and what physical law does it represent?
> [!faq]- Solution
> ∇ · J = 0
> It shows that steady electric currents are solenoidal (no sources/sinks) and represents Kirchhoff’s Current Law (KCL) in field theory (Σ I = 0).
Question 3: Faraday’s Law Integral Form
Write the integral form of Faraday’s Law of Electromagnetic Induction.
> [!faq]- Solution
> e = ∮_C E · dl = -d/dt ∫_S B · ds = -dΦ/dt
Question 4: Three Forms of Induced EMF
List the three physical mechanisms of induced EMF depending on whether the loop or magnetic field is moving/time-varying.
> [!faq]- Solution
> 1. Transformer EMF: Stationary loop in time-varying magnetic field (e_trans = -∫_S ∂B / ∂t · ds).
> 2. Motional EMF: Moving loop in static magnetic field (e_motion = ∮_C (u × B) · dl).
> 3. Combined EMF: Moving loop in time-varying magnetic field (e_total = e_trans + e_motion).
Question 5: Incompleteness of Ampere’s Static Law
Why is Ampere’s static curl equation ∇ × H = J mathematically incomplete for time-varying fields?
> [!faq]- Solution
> Taking the divergence yields ∇ · (∇ × H) = 0. But by continuity, ∇ · J = -∂ρ_v / ∂t ≠ 0 for dynamic fields, creating a mathematical contradiction unless displacement current J_d is added.
Question 6: Displacement Current Density Formula
Write the formula for displacement current density J_d in terms of electric flux density D and electric field E.
> [!faq]- Solution
> J_d = ∂D / ∂t = ε ∂E / ∂t [A/m^2]
Question 7: Maxwell’s Equations in Differential Form
List the four Maxwell’s equations in differential form for time-varying electromagnetic fields in a simple medium (ε, μ).
> [!faq]- Solution
> 1. ∇ · E = ρ_v / ε (Gauss’s Law for Electric Field)
> 2. ∇ · B = 0 (Gauss’s Law for Magnetic Field)
> 3. ∇ × E = -μ ∂H / ∂t (Faraday’s Law)
> 4. ∇ × H = J + ε ∂E / ∂t (Ampere-Maxwell Law)
Question 8: Poynting Vector Definition
Define the instantaneous Poynting Vector S and specify its SI units.
> [!faq]- Solution
> S = E × H [Watts/m^2]
> It represents the instantaneous directional power density carried by an electromagnetic wave.
Question 9: Poynting’s Theorem Physical Meaning
State the physical principle expressed by Poynting’s Theorem (∇ · S + ∂w / ∂t = -J · E).
> [!faq]- Solution
> The net outward power flux through a closed surface plus the rate of increase of stored electromagnetic energy inside equals the negative work done by the field on charges (ohmic power dissipation -J · E).
Question 10: Capacitor Conduction vs. Displacement Current
How does displacement current J_d ensure current continuity across a parallel-plate capacitor?
> [!faq]- Solution
> Inside the wires, physical conduction current I_c = dQ/dt flows. Between the insulated capacitor plates, J = 0, but time-varying electric flux ∂D / ∂t creates displacement current I_d = ∫ ∂D / ∂t ds = I_c, ensuring continuous total current across the circuit loop.