Formulate Maxwell Equations

Crack EM stuff. State right Maxwell equations (integral or differential). Apply boundary + symmetry to shrink system. Solve PDEs for fields. Compute Poynting vector, radiation pressure, wave impedance. Verify against static + wave limits.

When Use

• Boundary value problem for E + B fields, sources + material interfaces
• Derive EM wave equation from first principles
• Compute energy flow (Poynting vector) + momentum density of EM fields
• Apply boundary conditions at media interfaces (dielectric, conductor, magnetic)
• Analyze displacement current + role in Ampere-Maxwell
• Connect static limits (Coulomb, Biot-Savart) to unified time-dependent framework

Inputs

• Required: Physical setup (geometry, source charges + currents, material props)
• Required: Quantity to solve (E, B, wave solution, energy flux, boundary field)
• Optional: Symmetry (planar, cylindrical, spherical, none)
• Optional: Time dependence (static, harmonic omega, general)
• Optional: Boundary conditions at interfaces or conductor surfaces

Steps

Step 1: State Four Maxwell Equations + Pick Relevant Subset

Write full set, choose which constrain problem:

1. Gauss for E: div(E) = rho / epsilon_0 (diff) or closed_surface_integral(E . dA) = Q_enc / epsilon_0 (int). E divergence to charge density. Use for E from charge with symmetry.

2. Gauss for B: div(B) = 0 (diff) or closed_surface_integral(B . dA) = 0 (int). No magnetic monopoles. Every B line closed loop. Consistency check on B.

3. Faraday: curl(E) = -dB/dt (diff) or contour_integral(E . dl) = -d(Phi_B)/dt (int). Changing B makes curling E. Induction + wave derivation.

4. Ampere-Maxwell: curl(B) = mu_0 J + mu_0 epsilon_0 dE/dt (diff) or contour_integral(B . dl) = mu_0 I_enc + mu_0 epsilon_0 d(Phi_E)/dt (int). Current + changing E make curling B. Displacement current term mu_0 epsilon_0 dE/dt essential for wave + current continuity.

5. Form pick: Differential for local fields, wave equations, PDEs. Integral for high-symmetry where field extracts direct.

6. Active equations: Not all four independent every problem. Electrostatics (dB/dt = 0, J = 0) → only Gauss for E + curl(E) = 0 matter. Magnetostatics → Gauss for B + Ampere (no displacement current) enough.

## 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]

Got: Four equations stated. Relevant subset identified with justification. Displacement current included or explicitly argued negligible.

If fail: Unclear if displacement current matters? Estimate |epsilon_0 dE/dt| / |J|. Ratio near 1 or bigger → keep displacement current. Vacuum no free charges → always essential for wave.

Step 2: Apply Boundary Conditions + Symmetry

Shrink system with material interfaces + geometric symmetry:

1. Boundary at material interfaces: Medium 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 (continuous)
   • Normal B: B_1n = B_2n (continuous)
   • Tangential H: n_hat x (H_1 - H_2) = K_f (n_hat from 2 to 1)

2. Conductor boundary: Perfect conductor surface:

   • E_tangential = 0 (inside E = 0)
   • B_normal = 0 (inside B = 0 for time-varying)
   • Surface charge: sigma = epsilon_0 E_normal
   • Surface current: K = (1/mu_0) n_hat x B

3. Symmetry shrink: Use symmetries to cut independent variables:

   • Planar: fields depend one coord (z), PDEs → ODEs
   • Cylindrical: depend (rho, z) or rho only
   • Spherical: depend r only
   • Translational invariance: Fourier transform invariant direction

4. Gauge choice (using potentials): Pick gauge for scalar phi + vector A:

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

## 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]

Got: Boundary conditions stated every interface. Symmetry cuts dimension. Problem ready for PDE solution.

If fail: Over-determined (more equations than unknowns)? Check field components match conditions. Under-determined? Missed condition — often tangential H or radiation at infinity.

Step 3: Solve PDEs

Solve Maxwell equations or derived forms for field quantities:

1. Wave equation derive: Source-free, linear, homogeneous medium:

   • Curl of Faraday: curl(curl(E)) = -d/dt(curl(B))
   • Sub Ampere-Maxwell: curl(curl(E)) = -mu epsilon d^2E/dt^2
   • Vector identity: curl(curl(E)) = grad(div(E)) - nabla^2(E)
   • With div(E) = 0: nabla^2(E) = mu epsilon d^2E/dt^2
   • Wave speed: v = 1/sqrt(mu epsilon); vacuum c = 1/sqrt(mu_0 epsilon_0)
   • Same for B

2. Plane wave solutions: Wave in z-direction:

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

3. Laplace + Poisson (static):

   • No time: nabla^2(phi) = -rho/epsilon_0 (Poisson) or nabla^2(phi) = 0 (Laplace)
   • Separation of variables in right coordinates
   • Match boundary to pin expansion coefficients

4. Guided waves + cavities: Waveguides + resonant cavities:

   • Split into TE (transverse electric) + TM (transverse magnetic) modes
   • Apply conducting-wall boundary
   • Eigenvalue problem → allowed propagation constants + resonant frequencies
   • Cutoff: omega_c = v * pi * sqrt((m/a)^2 + (n/b)^2) for rectangular guide a x b

5. Skin depth in conductors: Time-varying fields into conductor conductivity sigma_c:

   • delta = sqrt(2 / (omega mu sigma_c))
   • Fields decay exp(-z/delta)
   • 60 Hz copper: delta ~ 8.5 mm; 1 GHz: delta ~ 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]

Got: Explicit field expressions satisfying Maxwell + all boundary. Dispersion relation or eigenvalue spectrum if applicable.

If fail: PDE won't separate in chosen coordinates? Try different system or numerical (finite difference, finite element). Solution fails a Maxwell equation on back-sub? Algebraic error — re-check curl + divergence.

Step 4: Compute Derived Quantities

Pull physical quantities from field solution:

1. Poynting vector: S = (1/mu_0) E x B (instantaneous energy flux, W/m^2):

   • Plane waves: S = (1/mu_0) |E|^2 / v in propagation direction
   • Time-averaged: <S> = (1/2) Re(E x H*) for harmonic
   • Intensity: I = |<S>| (power per area)

2. EM energy density:

   • u = (1/2)(epsilon_0 |E|^2 + |B|^2/mu_0) in vacuum
   • u = (1/2)(E . D + B . H) in linear media
   • Energy conservation: du/dt + div(S) = -J . E (Poynting's theorem)

3. Radiation pressure: Plane wave on surface:

   • Perfect absorber: P_rad = I/c = <S>/c
   • Perfect reflector: P_rad = 2I/c = 2<S>/c
   • Momentum flux density of EM field

4. Wave impedance:

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

5. Power dissipation + quality factor:

   • Ohmic loss per volume: p_loss = sigma |E|^2 / 2 (conductor)
   • Cavity Q-factor: Q = omega * (stored energy) / (power dissipated per cycle)
   • Bandwidth of resonances: 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]

Got: All derived quantities with right units. Energy conservation verified via Poynting's theorem. Magnitudes physically reasonable.

If fail: Poynting's theorem won't balance (du/dt + div(S) != -J . E)? Inconsistency E + B solutions. Re-verify both fields satisfy all four Maxwell. Common error: E + B from different approximations not mutually consistent.

Step 5: Verify Against Known Limits

Check solution reduces correctly in limits:

1. Static limit (omega -> 0): Solution → electrostatic or magnetostatic:

   • E satisfies Coulomb or Laplace/Poisson
   • B satisfies Biot-Savart or Ampere (no displacement current)
   • Displacement current vanishes: mu_0 epsilon_0 dE/dt -> 0

2. Plane wave limit: Source-free unbounded medium → plane waves, v = 1/sqrt(mu epsilon), correct polarization.

3. Perfect conductor limit (sigma -> infinity):

   • Skin depth delta -> 0 (no penetration)
   • Tangential E -> 0 at surface
   • Reflection r -> -1 (perfect reflection phase inversion)

4. Vacuum limit (epsilon_r = 1, mu_r = 1): Material-dependent → vacuum values. Wave speed = c. Impedance = eta_0 ~ 377 Ohms.

5. Energy conservation check: Integrate Poynting's theorem over closed volume. Rate of change of total field energy + power flowing out = negative of power from currents inside. Any imbalance = error.

## 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] |

Got: All limits produce correct known results. Energy conservation satisfied to numerical precision.

If fail: Failed limit = definite error. Static limit fail → source terms or boundary. Plane wave limit fail → wave equation derivation. Energy conservation fail → inconsistency E + B. Trace failure to step, fix before accepting.

Checks

• All four Maxwell equations stated + relevant subset identified
• Displacement current included or explicitly justified negligible
• Boundary conditions applied at every material interface
• Symmetry shrinks PDE dimension
• Wave equation (or Laplace/Poisson) correctly derived
• Field solutions satisfy all four Maxwell on back-sub
• Poynting vector + energy density with right units (W/m^2 + J/m^3)
• Poynting's theorem (energy conservation) verified
• Wave impedance + reflection/transmission coefficients reasonable
• Static limit reproduces Coulomb + Biot-Savart
• Plane wave limit gives v = 1/sqrt(mu epsilon) + orthogonal E, B, k
• Solution complete enough for reproduction

Pitfalls

• Dropping displacement current: Original Ampere (curl B = mu_0 J), divergence gives div(J) = 0, contradicts charge conservation when rho changes in time. Term mu_0 epsilon_0 dE/dt fixes, essential for wave propagation. Never drop without verifying dE/dt negligible vs J/epsilon_0.
• Inconsistent E + B solutions: Solving E + B independent (E from Gauss, B from Ampere) without verifying Faraday + Gauss for B → fields not mutually consistent. Always verify all four.
• Wrong boundary normal direction: Convention n_hat x (H_1 - H_2) = K_f needs n_hat from medium 2 into 1. Reversed → flips surface current sign.
• Confusing D, E, B, H in materials: Vacuum: D = epsilon_0 E + B = mu_0 H. Linear media: D = epsilon E + B = mu H. Maxwell in matter use D + H for free source, E + B for force law. Mixing → factor of epsilon_r or mu_r errors.
• Phase velocity vs group velocity: Wave speed v = omega/k = phase velocity. Energy + info propagate at group velocity v_g = d(omega)/dk. Dispersive media differ, phase for energy transport → wrong.
• Forgetting radiation condition: Scattering + radiation in unbounded domain → solution must satisfy Sommerfeld radiation condition (outgoing waves at infinity). Without it → not unique, may include unphysical incoming waves.

See Also

• analyze-magnetic-field -- compute static B-fields = magnetostatic limit of Maxwell
• solve-electromagnetic-induction -- apply Faraday to induction geometries + RL circuits
• formulate-quantum-problem -- quantize EM field for quantum optics + QED
• derive-theoretical-result -- rigorous derivation of wave equations, Green's functions, dispersion
• analyze-diffusion-dynamics -- diffusion equations from Maxwell in conducting media (skin effect)