Formulate Maxwell Equations

EM phenomenon → state Maxwell eqs (int/diff) → apply BCs + symmetry → solve PDE → derive Poynting/pressure/impedance → verify limits.

Use When

• BVP for E, B w/ sources + interfaces
• Derive EM wave eq from first principles
• Compute Poynting vector + momentum density
• BCs at interfaces (dielectrics, conductors, magnetic)
• Displacement current role in Ampere-Maxwell
• Static limits (Coulomb, Biot-Savart) → unified time-dep

In

• Required: config (geometry, sources, material props)
• Required: target quantity (E, B, wave, flux, boundary value)
• Optional: symmetry (planar/cyl/sph/none)
• Optional: time-dep (static/harmonic omega/general)
• Optional: BCs at interfaces/conductors

Do

Step 1: State 4 eqs + select subset

1. Gauss E: div(E)=rho/epsilon_0 (diff) or ∮E·dA=Q_enc/epsilon_0 (int). Use w/ symmetry for E from charges.

2. Gauss B: div(B)=0 (diff) or ∮B·dA=0 (int). No monopoles. Consistency check.

3. Faraday: curl(E)=-dB/dt (diff) or ∮E·dl=-dPhi_B/dt (int). Changing B → curling E. Induction + waves.

4. Ampere-Maxwell: curl(B)=mu_0 J + mu_0 epsilon_0 dE/dt (diff) or ∮B·dl=mu_0 I_enc + mu_0 epsilon_0 dPhi_E/dt (int). Displacement current mu_0 epsilon_0 dE/dt → essential for waves + continuity.

5. Form: diff for local/wave/PDE; int for symmetry problems.

6. Active subset: electrostatics → Gauss E + curl(E)=0. Magnetostatics → Gauss B + Ampere (no displacement).

## Maxwell Equations for This Problem
- **Form**: [differential / integral / both]
- **Active equations**: [list which of the four are non-trivial constraints]
- **Source terms**: rho = [charge density], J = [current density]
- **Time dependence**: [static / harmonic / general]
- **Displacement current**: [negligible / essential -- with justification]

→ 4 eqs stated, subset ID'd w/ justify, displacement current included/argued negligible.

If err: unclear displacement current → estimate |epsilon_0 dE/dt|/|J|. ≥1 → keep. Vacuum + no free charges → always essential.

Step 2: BCs + symmetry

1. BCs at interfaces (media 1,2, surface charge sigma_f, surface current K_f):

   • Normal E: epsilon_1 E_1n - epsilon_2 E_2n = sigma_f
   • Tangential E: E_1t = E_2t
   • Normal B: B_1n = B_2n
   • Tangential H: n_hat × (H_1-H_2) = K_f (n_hat: 2→1)

2. Conductor BCs (perfect):

   • E_tan=0 (inside E=0)
   • B_normal=0 (time-varying)
   • sigma = epsilon_0 E_normal
   • K = (1/mu_0) n_hat × B

3. Symmetry:

   • Planar: fields depend on 1 coord → ODE
   • Cylindrical: (rho,z) or rho
   • Spherical: r only
   • Translational invariance → Fourier transform

4. Gauge (potentials phi, A):

   • Coulomb: div(A)=0 (separates static + radiation)
   • Lorenz: div(A) + mu_0 epsilon_0 d(phi)/dt=0 (Lorentz-covariant, decouples wave eqs)

## Boundary Conditions and Symmetry
- **Interfaces**: [list with media properties on each side]
- **Boundary conditions applied**: [normal E, tangential E, normal B, tangential H]
- **Symmetry**: [planar / cylindrical / spherical / none]
- **Reduced coordinates**: [independent variables after symmetry reduction]
- **Gauge** (if using potentials): [Coulomb / Lorenz / other]

→ BCs at every interface, symmetry applied, ready for PDE.

If err: over-determined → check components vs conditions. Under-determined → missed BC (often tangential H or radiation at ∞).

Step 3: Solve PDE

1. Wave eq derivation (source-free, linear, homogeneous):

   • curl(curl(E)) = -d/dt(curl(B)) = -mu epsilon d²E/dt²
   • Identity: curl(curl(E)) = grad(div(E)) - nabla²(E)
   • div(E)=0 → nabla²(E) = mu epsilon d²E/dt²
   • v=1/sqrt(mu epsilon); vacuum c=1/sqrt(mu_0 epsilon_0)
   • Same for B

2. Plane wave (z-prop):

   • E(z,t)=E_0 exp[i(kz - omega t)], k=omega/v=omega*sqrt(mu epsilon)
   • B=(1/v) k_hat × E
   • |B|=|E|/v
   • Polarization: linear/circular/elliptical

3. Laplace/Poisson (static):

   • nabla²(phi)=-rho/epsilon_0 (Poisson) or nabla²(phi)=0 (Laplace)
   • Separation of variables → match BCs

4. Guided waves/cavities:

   • TE/TM decomposition
   • Conducting-wall BCs
   • Eigenvalue → propagation const / resonance
   • Cutoff: omega_c = vpisqrt((m/a)²+(n/b)²) rect guide a×b

5. Skin depth:

   • delta = sqrt(2/(omega mu sigma_c))
   • Decay exp(-z/delta)
   • 60 Hz Cu: ~8.5 mm; 1 GHz: ~2 micrometers

## Field Solution
- **Equation solved**: [wave equation / Laplace / Poisson / eigenvalue]
- **Solution method**: [separation of variables / Fourier transform / Green's function / numerical]
- **Result**: E(r, t) = [expression], B(r, t) = [expression]
- **Dispersion relation**: omega(k) = [if wave solution]
- **Characteristic scales**: [wavelength, skin depth, decay length]

→ Explicit E, B satisfying all eqs + BCs, dispersion/eigenvalues if applicable.

If err: can't separate → try new coord or numerics (FD/FE). Back-sub fails Maxwell → algebra err in curl/div.

Step 4: Derived quantities

1. Poynting: S = (1/mu_0) E × B (W/m²)

   • Plane wave: S = (1/mu_0) |E|²/v in prop dir
   • Time-avg: <S> = (1/2) Re(E × H*) harmonic
   • Intensity: I = |<S>|

2. Energy density:

   • u = (1/2)(epsilon_0 |E|² + |B|²/mu_0) vacuum
   • u = (1/2)(E·D + B·H) linear media
   • Conservation: du/dt + div(S) = -J·E (Poynting's thm)

3. Radiation pressure:

   • Absorber: P_rad = I/c = <S>/c
   • Reflector: P_rad = 2I/c = 2<S>/c

4. Wave impedance:

   • Medium: eta = sqrt(mu/epsilon) = mu*v
   • Vacuum: eta_0 = sqrt(mu_0/epsilon_0) ≈ 377 Ω
   • |E| = eta |H|
   • Reflection normal: r = (eta_2 - eta_1)/(eta_2 + eta_1)

5. Power + Q:

   • Ohmic loss/vol: p_loss = sigma |E|²/2
   • Q = omega * (stored energy)/(power dissipated/cycle)
   • Bandwidth: Delta_omega = omega/Q

## Derived Quantities
- **Poynting vector**: S = [expression], <S> = [time-averaged]
- **Energy density**: u = [expression]
- **Radiation pressure**: P_rad = [value]
- **Wave impedance**: eta = [value]
- **Reflection/transmission**: r = [value], t = [value]
- **Q-factor** (if resonant): Q = [value]

→ All quantities w/ correct units, energy conservation via Poynting, reasonable magnitudes.

If err: Poynting thm not balanced → E/B inconsistent. Re-verify all 4 eqs. Common: E, B from different approx not mutually consistent.

Step 5: Verify limits

1. Static (omega→0):

   • E → Coulomb / Laplace-Poisson
   • B → Biot-Savart / Ampere (no displacement)
   • Displacement → 0

2. Plane wave: v=1/sqrt(mu epsilon), correct polarization.

3. Perfect conductor (sigma→∞):

   • delta → 0
   • E_tan → 0 at surface
   • r → -1 (phase inversion)

4. Vacuum (epsilon_r=1, mu_r=1): v=c, eta=eta_0 ≈ 377 Ω.

5. Energy conservation: integrate Poynting thm over closed vol → total field energy rate + outflow = -power from currents. Imbalance = err.

## Limiting Case Verification
| Limit | Condition | Expected | Obtained | Match |
|-------|-----------|----------|----------|-------|
| Static | omega -> 0 | Coulomb / Biot-Savart | [result] | [Yes/No] |
| Plane wave | unbounded medium | v = c/n, eta = eta_0/n | [result] | [Yes/No] |
| Perfect conductor | sigma -> inf | delta -> 0, r -> -1 | [result] | [Yes/No] |
| Vacuum | epsilon_r = mu_r = 1 | c, eta_0 | [result] | [Yes/No] |
| Energy conservation | Poynting's theorem | balanced | [check] | [Yes/No] |

→ All limits match. Energy conserved to numerical precision.

If err: Static fail → source/BC err. Plane wave fail → wave eq derivation err. Energy conservation fail → E/B inconsistent. Trace + fix before accepting.

Check

• 4 eqs stated, subset ID'd
• Displacement current included or justified negligible
• BCs applied at every interface
• Symmetry reduces PDE dim
• Wave eq / Laplace / Poisson correctly derived
• Solutions back-sub satisfy all 4 eqs
• Poynting + energy density correct units (W/m², J/m³)
• Poynting thm verified
• Impedance + r, t reasonable
• Static limit = Coulomb + Biot-Savart
• Plane wave limit: v=1/sqrt(mu epsilon), E⊥B⊥k
• Solution reproducible

Traps

• Drop displacement: div(curl B)=0 → div(J)=0 contradicts charge conservation → mu_0 epsilon_0 dE/dt essential. Never drop w/o checking dE/dt vs J/epsilon_0.
• Inconsistent E, B: solving independently can violate Faraday + Gauss B. Always verify all 4.
• Wrong n_hat dir: n_hat × (H_1-H_2)=K_f requires 2→1. Reverse flips sign.
• D/E/B/H confusion: vacuum D=epsilon_0 E, B=mu_0 H. Media D=epsilon E, B=mu H. Maxwell uses D,H for free sources, E,B for force. Mixing → epsilon_r/mu_r errors.
• Phase vs group v: v=omega/k phase. Energy/info → v_g=d(omega)/dk. Dispersive media: differ.
• Forget radiation condition: scattering in unbounded → Sommerfeld (outgoing at ∞). Missing → non-unique + unphysical incoming.

→

• analyze-magnetic-field — static B (magnetostatic limit)
• solve-electromagnetic-induction — Faraday + RL circuits
• formulate-quantum-problem — quantize EM (QED)
• derive-theoretical-result — rigorous wave/Green's/dispersion
• analyze-diffusion-dynamics — diffusion eq from Maxwell (skin effect)