Problem. The electro-magnetic field in a microwave oven is given by Maxwell's equations: V × E = -μHt V x H = ЄEt +σE, where the vector E is the electric field and the vector H is the magnetic field. The constants ε, σ and μ are the permittivity, the electrical conduction and the permeability. In the air, one has ₁ = 8.85. 10-12, 0₁ = 0 and μ₁ = 4л 10-7. In the chicken, one has εp = 4.43 · 10-11, 3. 10-11 and Мр = 4π-10-7. (1) We assume that the solution to the above PDE is time-harmonic: ,0p = with w = E=E(x, y, z)eit, H = H(x, y, z)eiwt, 2πf=22.45 109. In this case, show that Maxwell's equations then read V × E = -iwμH V x H = iw E, with to be determined. (2) We assume that E = 0₂H = 0. Find a new equation for the third component of the electric field E3. This equation is Helmoltz's equation in 2d: AЕ3+w² μЕ3 = 0.

Problem. The electro-magnetic field in a microwave oven is given by Maxwell's equations: V × E = -μHt V x H = ЄEt +σE, where the vector E is the electric field and the vector H is the magnetic field. The constants ε, σ and μ are the permittivity, the electrical conduction and the permeability. In the air, one has ₁ = 8.85. 10-12, 0₁ = 0 and μ₁ = 4л 10-7. In the chicken, one has εp = 4.43 · 10-11, 3. 10-11 and Мр = 4π-10-7. (1) We assume that the solution to the above PDE is time-harmonic: ,0p = with w = E=E(x, y, z)eit, H = H(x, y, z)eiwt, 2πf=22.45 109. In this case, show that Maxwell's equations then read V × E = -iwμH V x H = iw E, with to be determined. (2) We assume that E = 0₂H = 0. Find a new equation for the third component of the electric field E3. This equation is Helmoltz's equation in 2d: AЕ3+w² μЕ3 = 0.