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Solves electromagnetic boundary value problems using Maxwell's equations. Derives wave equations, computes Poynting vector, applies boundary conditions at interfaces.
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Analyze electromagnetic phenomena by stating the relevant Maxwell equations in appropriate form (integral or differential), applying boundary conditions and symmetry to reduce the system, solving the resulting partial differential equations for the fields, computing derived quantities such as the Poynting vector, radiation pressure, and wave impedance, and verifying the solution against known s...
Analyze electromagnetic phenomena by stating the relevant Maxwell equations in appropriate form (integral or differential), applying boundary conditions and symmetry to reduce the system, solving the resulting partial differential equations for the fields, computing derived quantities such as the Poynting vector, radiation pressure, and wave impedance, and verifying the solution against known static and wave limits.
Write the complete set and select which equations constrain the problem:
Gauss's law for E: div(E) = rho / epsilon_0 (differential) or closed_surface_integral(E . dA) = Q_enc / epsilon_0 (integral). Relates E-field divergence to charge density. Use for finding E from charge distributions with symmetry.
Gauss's law for B: div(B) = 0 (differential) or closed_surface_integral(B . dA) = 0 (integral). No magnetic monopoles. Every magnetic field line is a closed loop. Use as a consistency check on computed B-fields.
Faraday's law: curl(E) = -dB/dt (differential) or contour_integral(E . dl) = -d(Phi_B)/dt (integral). A changing B-field generates a curling E-field. Use for induction problems and wave derivation.
Ampere-Maxwell law: curl(B) = mu_0 J + mu_0 epsilon_0 dE/dt (differential) or contour_integral(B . dl) = mu_0 I_enc + mu_0 epsilon_0 d(Phi_E)/dt (integral). Current and changing E-field generate curling B-field. The displacement current term mu_0 epsilon_0 dE/dt is essential for wave propagation and current continuity.
Form selection: Choose differential form for local field calculations, wave equations, and PDEs. Choose integral form for high-symmetry problems where the field can be extracted from the integral directly.
Identify active equations: Not all four equations are independent constraints in every problem. For electrostatics (dB/dt = 0, J = 0), only Gauss's law for E and curl(E) = 0 matter. For magnetostatics, Gauss's law for B and Ampere's law (without displacement current) suffice.
## 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]
Expected: The four equations are stated, the relevant subset is identified with justification, and the displacement current is either included or explicitly argued to be negligible.
On failure: If it is unclear whether the displacement current matters, estimate the ratio |epsilon_0 dE/dt| / |J|. If this ratio is comparable to or greater than 1, the displacement current must be retained. In vacuum with no free charges, the displacement current is always essential for wave propagation.
Reduce the system using material interfaces and geometric symmetry:
Boundary conditions at material interfaces: At the interface between media 1 and 2 with surface charge sigma_f and surface current K_f:
Conductor boundary conditions: At the surface of a perfect conductor:
Symmetry reduction: Use identified symmetries to reduce the number of independent variables:
Gauge choice (if using potentials): Select a gauge for the scalar potential phi and vector potential A:
## 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]
Expected: All boundary conditions are stated at every interface, symmetry is exploited to reduce the dimensionality, and the problem is ready for PDE solution.
On failure: If boundary conditions are over-determined (more equations than unknowns at an interface), check that the number of field components matches the number of conditions. If under-determined, a boundary condition has been missed -- often the tangential H condition or the radiation condition at infinity.
Solve the Maxwell equations or their derived forms for the field quantities:
Wave equation derivation: In a source-free, linear, homogeneous medium:
Plane wave solutions: For a wave propagating in the z-direction:
Laplace and Poisson equations (static case):
Guided waves and cavities: For waveguides and resonant cavities:
Skin depth in conductors: For time-varying fields penetrating a conductor with conductivity sigma_c:
## 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]
Expected: Explicit field expressions satisfying Maxwell's equations and all boundary conditions, with the dispersion relation or eigenvalue spectrum if applicable.
On failure: If the PDE cannot be separated in the chosen coordinate system, try a different system or resort to numerical methods (finite difference, finite element). If the solution does not satisfy one of the Maxwell equations on back-substitution, there is an algebraic error in the derivation -- re-check the curl and divergence operations.
Extract physically meaningful quantities from the field solution:
Poynting vector: S = (1/mu_0) E x B (instantaneous energy flux, W/m^2):
Electromagnetic energy density:
Radiation pressure: For a plane wave incident on a surface:
Wave impedance:
Power dissipation and quality factor:
## 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]
Expected: All derived quantities computed with correct units, energy conservation verified via Poynting's theorem, and physically reasonable magnitudes.
On failure: If Poynting's theorem does not balance (du/dt + div(S) does not equal -J . E), there is an inconsistency between the E and B solutions. Re-verify that both fields satisfy all four Maxwell equations simultaneously. A common error is computing E and B from different approximations that are not mutually consistent.
Check that the full solution reduces correctly in limiting cases:
Static limit (omega -> 0): The solution should reduce to the electrostatic or magnetostatic result:
Plane wave limit: In a source-free, unbounded medium, the solution should reduce to plane waves with v = 1/sqrt(mu epsilon) and the correct polarization.
Perfect conductor limit (sigma -> infinity):
Vacuum limit (epsilon_r = 1, mu_r = 1): Material-dependent quantities should reduce to their vacuum values. Wave speed should equal c. Impedance should equal eta_0 approximately 377 Ohms.
Energy conservation check: Integrate Poynting's theorem over a closed volume. The rate of change of total field energy plus the power flowing out through the surface must equal the negative of the power delivered by currents inside the volume. Any imbalance indicates an 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] |
Expected: All limits produce the correct known results. Energy conservation is satisfied to within numerical precision.
On failure: A failed limit is a definitive indicator of an error. The static limit failing suggests a problem in the source terms or boundary conditions. The plane wave limit failing suggests an error in the wave equation derivation. Energy conservation failing suggests inconsistency between E and B solutions. Trace the failure back to the specific step and correct before accepting the solution.
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Solves mechanics, fluid dynamics, and electromagnetism problems by applying conservation of energy, momentum, angular momentum, or charge — identifying the system boundary, the conserved quantity, and conservation conditions.
Performs symbolic mathematics in Python with SymPy: solve equations algebraically, compute derivatives/integrals/limits, manipulate expressions, work with symbolic matrices, and generate code from formulas. Use when exact symbolic results are needed.