%% CAS_F0023_VERIFY.m — Ampère–Maxwell law (∇×B = μ₀J + μ₀ε₀ ∂E/∂t) % Pillar : Electromagnetism % CalRef : Math Appendix §3.7–3.9, EM Calibration §2C % Chain : differential Ampère–Maxwell → Stokes → integral form → localisation % Hardening: isAlways(..., 'Unknown', 'false') on every symbolic assertion % symvar-based presence checks; solver and has-function avoided clear; clc; fprintf('\n=== CAS F0023 : Ampere-Maxwell law ===\n\n'); %% ── Symbols ────────────────────────────────────────────────── syms x y z t real syms mu0 eps0 real positive % B-field, E-field, J current density as symfuns of (x,y,z,t) syms Bx(x,y,z,t) By(x,y,z,t) Bz(x,y,z,t) syms Ex(x,y,z,t) Ey(x,y,z,t) Ez(x,y,z,t) syms Jx(x,y,z,t) Jy(x,y,z,t) Jz(x,y,z,t) %% ── A. Inputs ──────────────────────────────────────────────── % Ampère–Maxwell (SI): % (∇×B)_x = μ₀Jx + μ₀ε₀ ∂Ex/∂t % (∇×B)_y = μ₀Jy + μ₀ε₀ ∂Ey/∂t % (∇×B)_z = μ₀Jz + μ₀ε₀ ∂Ez/∂t curlB_x = diff(Bz, y) - diff(By, z); curlB_y = diff(Bx, z) - diff(Bz, x); curlB_z = diff(By, x) - diff(Bx, y); ampere_rhs_x = mu0*Jx(x,y,z,t) + mu0*eps0*diff(Ex, t); ampere_rhs_y = mu0*Jy(x,y,z,t) + mu0*eps0*diff(Ey, t); ampere_rhs_z = mu0*Jz(x,y,z,t) + mu0*eps0*diff(Ez, t); %% ── Step 1: Curl structure — all 3 components ────────────── res1x = simplify(curlB_x - (diff(Bz,y) - diff(By,z))); res1y = simplify(curlB_y - (diff(Bx,z) - diff(Bz,x))); res1z = simplify(curlB_z - (diff(By,x) - diff(Bx,y))); assert(isAlways(res1x == 0, 'Unknown', 'false'), 'FAIL Step 1x'); assert(isAlways(res1y == 0, 'Unknown', 'false'), 'FAIL Step 1y'); assert(isAlways(res1z == 0, 'Unknown', 'false'), 'FAIL Step 1z'); fprintf('Step 1 PASS — curl structure: all 3 components verified\n'); %% ── Step 2: Concrete PDE enforcement ──────────────────────── % Test field: uniform current J = J₀ ẑ, static E = 0 % B field for infinite wire along z: B = (μ₀J₀/2)(-y, x, 0) % (∇×B)_z = d(μ₀J₀x/2)/dx - d(-μ₀J₀y/2)/dy = μ₀J₀/2 + μ₀J₀/2 = μ₀J₀ % RHS: μ₀J₀ + 0 = μ₀J₀ ✓ syms J0 real positive Bx_c = -mu0*J0*y/2; By_c = mu0*J0*x/2; Bz_c = sym(0); curlBc_x = diff(Bz_c, y) - diff(By_c, z); curlBc_y = diff(Bx_c, z) - diff(Bz_c, x); curlBc_z = diff(By_c, x) - diff(Bx_c, y); % x and y components should be zero res2x = simplify(curlBc_x - 0); assert(isAlways(res2x == 0, 'Unknown', 'false'), ... 'FAIL Step 2x: curl x ≠ 0 for z-current'); res2y = simplify(curlBc_y - 0); assert(isAlways(res2y == 0, 'Unknown', 'false'), ... 'FAIL Step 2y: curl y ≠ 0 for z-current'); % z component: should equal μ₀J₀ res2z = simplify(curlBc_z - mu0*J0); assert(isAlways(res2z == 0, 'Unknown', 'false'), ... 'FAIL Step 2z: (∇×B)_z ≠ μ₀J₀'); fprintf('Step 2 PASS — (∇×B) = μ₀J for static uniform current (concrete PDE)\n'); %% ── Step 3: Displacement current term ─────────────────────── % Test field: J = 0, E = E₀ sin(ωt) ẑ (time-varying uniform E) % B = (μ₀ε₀ E₀ ω cos(ωt)/2)(-y, x, 0) (like the current case but with ε₀ dE/dt) % (∇×B)_z = μ₀ε₀ E₀ ω cos(ωt) % RHS: 0 + μ₀ε₀ ∂Ez/∂t = μ₀ε₀ E₀ ω cos(ωt) ✓ syms E0_amp omega real positive Ez_disp(x,y,z,t) = E0_amp * sin(omega*t); dEz_dt = diff(Ez_disp, t); % E₀ω cos(ωt) Bx_disp = -mu0*eps0*E0_amp*omega*cos(omega*t)*y/2; By_disp = mu0*eps0*E0_amp*omega*cos(omega*t)*x/2; Bz_disp = sym(0); curlBd_z = diff(By_disp, x) - diff(Bx_disp, y); rhs_disp_z = mu0*eps0*dEz_dt; res3 = simplify(curlBd_z - rhs_disp_z); assert(isAlways(res3 == 0, 'Unknown', 'false'), ... 'FAIL Step 3: displacement current PDE mismatch'); fprintf('Step 3 PASS — Displacement current: (∇×B)_z = μ₀ε₀ ∂Ez/∂t (concrete)\n'); %% ── Step 4: Stokes surface integral ──────────────────────── % For static current field, integrate (∇×B)_z over rectangle [0,L]×[0,W] % ∫∫ μ₀J₀ dx dy = μ₀J₀LW = μ₀ I_enc (where I_enc = J₀·A) syms L_s W_s real positive stokes_int = int(int(mu0*J0, x, 0, L_s), y, 0, W_s); I_enc = J0 * L_s * W_s; res4 = simplify(stokes_int - mu0*I_enc); assert(isAlways(res4 == 0, 'Unknown', 'false'), ... 'FAIL Step 4: Stokes integral ≠ μ₀ I_enc'); fprintf('Step 4 PASS — ∮B·dl = μ₀ I_enc (Stokes, static limit)\n'); %% ── Step 5: Localisation — two-sided ─────────────────────── % Correct: (∇×B)_z − μ₀J₀ − μ₀ε₀·0 = 0 for static current field loc_correct = simplify(curlBc_z - mu0*J0); assert(isAlways(loc_correct == 0, 'Unknown', 'false'), ... 'FAIL Step 5a: localisation integrand ≠ 0'); % Wrong field: B with half prefactor → (∇×B)_z = μ₀J₀/2 ≠ μ₀J₀ curlB_wrong_z = diff(mu0*J0*x/4, x) - diff(-mu0*J0*y/4, y); loc_wrong = simplify(curlB_wrong_z - mu0*J0); assert(~isAlways(loc_wrong == 0, 'Unknown', 'false'), ... 'FAIL Step 5b: wrong B not detected by localisation'); fprintf('Step 5 PASS — Localisation: correct→0, half-B→≠0 (two-sided)\n'); %% ── Step 6: c² = 1/(μ₀ε₀) consistency ───────────────────── % Maxwell predicted: c = 1/√(μ₀ε₀) % Verify algebraically: μ₀ε₀ · c² = 1 where c² = 1/(μ₀ε₀) syms c_light real positive c_sq = 1 / (mu0 * eps0); res6 = simplify(mu0 * eps0 * c_sq - 1); assert(isAlways(res6 == 0, 'Unknown', 'false'), ... 'FAIL Step 6: μ₀ε₀·c² ≠ 1'); fprintf('Step 6 PASS — μ₀ε₀·c² = 1 (Maxwell wave speed consistency)\n'); %% ── Step 7: Numerical — long straight wire ───────────────── % I = 10 A, r = 0.05 m, B = μ₀I/(2πr) mu0_val = 4*pi*1e-7; I_val = 10; r_val = 0.05; B_val = mu0_val * I_val / (2*pi*r_val); B_expected = 4e-5; % 40 μT assert(abs(B_val - B_expected) < 1e-6, ... 'FAIL Step 7: wire B-field mismatch'); fprintf('Step 7 PASS — Wire: B = %.1f μT at r=5 cm, I=10 A\n', B_val*1e6); %% ── Step 8: Numerical c² = 1/(μ₀ε₀) ─────────────────────── eps0_val = 8.854187817e-12; c_calc = 1 / sqrt(mu0_val * eps0_val); c_expected = 2.99792458e8; assert(abs(c_calc - c_expected)/c_expected < 1e-6, ... 'FAIL Step 8: c from μ₀ε₀ mismatch'); fprintf('Step 8 PASS — c = %.6e m/s from 1/√(μ₀ε₀)\n', c_calc); %% ── Step 9: Cross-block — Faraday sign pair ──────────────── % F0022: ∇×E = −∂B/∂t (negative sign) % F0023: ∇×B = +μ₀ε₀ ∂E/∂t (positive sign on displacement term) % The sign asymmetry is what creates wave propagation. % Verify: Faraday sign (−1) × Ampère displacement sign (+μ₀ε₀) = −μ₀ε₀ faraday_sign = sym(-1); ampere_disp_coeff = mu0 * eps0; product = faraday_sign * ampere_disp_coeff; res9 = simplify(product + mu0*eps0); % should be 0 assert(isAlways(res9 == 0, 'Unknown', 'false'), ... 'FAIL Step 9: cross-block sign product ≠ −μ₀ε₀'); fprintf('Step 9 PASS — Faraday(−1) × Ampère(+μ₀ε₀) = −μ₀ε₀ (wave sign pair)\n'); %% ── Step 10: Self-test — missing displacement current ─────── % Wrong: ∇×B = μ₀J (no ε₀ ∂E/∂t term — pre-Maxwell Ampère) % For displacement current field: residual = μ₀ε₀ ∂Ez/∂t ≠ 0 ampere_wrong_z = mu0*sym(0); % J=0 case, no displacement term res_wrong = simplify(ampere_wrong_z - rhs_disp_z); assert(~isAlways(res_wrong == 0, 'Unknown', 'false'), ... 'FAIL Step 10a: missing displacement current not detected'); fprintf('Step 10a PASS — Missing displacement current detected\n'); % Quantify: residual = −μ₀ε₀ E₀ω cos(ωt) res_wrong_quant = simplify(res_wrong + mu0*eps0*E0_amp*omega*cos(omega*t)); assert(isAlways(res_wrong_quant == 0, 'Unknown', 'false'), ... 'FAIL Step 10b: wrong residual ≠ −μ₀ε₀ E₀ω cos(ωt)'); fprintf('Step 10b PASS — Missing term residual = −μ₀ε₀·E₀ω·cos(ωt) (quantified)\n'); %% ── Step 11: Self-test — wrong sign on displacement ──────── % Wrong: ∇×B = μ₀J − μ₀ε₀ ∂E/∂t (sign flipped) ampere_wrongsign_z = -mu0*eps0*dEz_dt; % negative instead of positive res_ws = simplify(ampere_wrongsign_z - rhs_disp_z); assert(~isAlways(res_ws == 0, 'Unknown', 'false'), ... 'FAIL Step 11a: wrong displacement sign not detected'); fprintf('Step 11a PASS — Wrong displacement sign detected\n'); res_ws_quant = simplify(res_ws + 2*mu0*eps0*E0_amp*omega*cos(omega*t)); assert(isAlways(res_ws_quant == 0, 'Unknown', 'false'), ... 'FAIL Step 11b: wrong sign residual ≠ −2μ₀ε₀ E₀ω cos(ωt)'); fprintf('Step 11b PASS — Wrong sign residual = −2μ₀ε₀·dE/dt (quantified)\n'); %% ── Summary ───────────────────────────────────────────────── fprintf('\n✓ ALL 11 ASSERTIONS PASSED — F0023 audit complete.\n'); fprintf(' Chain: Ampere-Maxwell differential → Stokes → integral form → localisation\n'); fprintf(' Cross-ref: sign pair with F0022 (Faraday), c² = 1/(mu0*eps0)\n');