- 2076 Asoj · 10 marks
Maxwell's equations, displacement current
Short answer
Maxwell's equations: div D = rho_v, integral D·dS = Q; div B = 0, integral B·dS = 0; curl E = -dB/dt, integral E·dl = -d/dt integral B·dS; curl H = J + dD/dt, integral H·dl = integral (J + dD/dt)·dS. Displacement current J_d = dD/dt ensures continuity. Depth delta = sqrt(2/(omega mu sigma)), distance where field falls to 1/e.
State and explain Maxwell's equations in differential and integral form. Also define displacement current and depth of penetration.
Maxwell's equations
Maxwell's four equations are the complete set of laws of classical electromagnetism. They combine Gauss's law (electric), Gauss's law (magnetic), Faraday's law and Ampere's circuital law (corrected with displacement current).
| Law | Differential (point) form | Integral form |
|---|---|---|
| Gauss's law (electric) | ∇·D = ρv | ∮ D·dS = ∫ ρv dv = Q |
| Gauss's law (magnetic) | ∇·B = 0 | ∮ B·dS = 0 |
| Faraday's law | ∇×E = −∂B/∂t | ∮ E·dl = −d/dt ∫ B·dS |
| Ampere's law | ∇×H = J + ∂D/∂t | ∮ H·dl = ∫ (J + ∂D/∂t)·dS |
Explanation of each:
- Electric Gauss law: electric flux leaving a closed surface equals the free charge enclosed. Charge is the source of D.
- Magnetic Gauss law: there are no magnetic monopoles. Magnetic flux lines are closed loops, so net B flux through any closed surface is zero.
- Faraday's law: a time-varying magnetic flux induces an emf (a circulating E field). The minus sign is Lenz's law.
- Ampere's law: a magnetic field circulates around conduction current J and around a changing electric flux (displacement current ∂D/∂t).
Supporting relations: D = εE, B = μH, J = σE. For a static field, the time derivatives vanish and the equations reduce to the static laws.
Displacement current
Ampere's law in the form ∇×H = J fails for time-varying fields because ∇·(∇×H) = 0 would force ∇·J = 0, which contradicts the continuity equation ∇·J = −∂ρv/∂t. Maxwell added the term
J_d = ∂D/∂t (A/m²)
so that ∇×H = J + J_d. Displacement current is the current "through" a dielectric or vacuum due to a changing electric flux density. It is not a flow of charges, but it produces a magnetic field like a real current. It makes current continuous through a capacitor and is the reason electromagnetic waves can exist in free space.
Depth of penetration (skin depth)
When a wave enters a conducting medium, its amplitude decays as e^(−αz). The depth of penetration δ is the distance at which the amplitude falls to 1/e (about 36.8 %) of its surface value:
δ = 1/α
For a good conductor (σ ≫ ωε), α = β = √(πfμσ), so
δ = √(2/(ωμσ)) = 1/√(πfμσ)
A higher frequency or higher conductivity gives a smaller δ, so the fields and currents stay near the surface (skin effect). Example: copper at 1 MHz has δ ≈ 66 μm.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- All Maxwell equations, point and integral · Electromagnetism, chapter 3
- Maxwell's equations in phasor form · Electromagnetism, chapter 3 · asked 2×
- Faraday's law and modified Maxwell equations · Electromagnetism, chapter 3
- Displacement vs conduction current · Electromagnetism, chapter 3 · asked 2×
- Displacement current density in transformer · Electromagnetism, chapter 3
- Conduction and displacement current in capacitor · Electromagnetism, chapter 3
- Maxwell's equations, integral to differential · Engineering Physics, chapter 9 · asked 3×
- EM wave in dielectric from Maxwell's equations · Engineering Physics, chapter 9 · asked 2×
- Maxwell's equations and continuity equation · Engineering Physics, chapter 9 · asked 2×
- Significance of differential equations; liver model · Simulation and Modeling, chapter 3