Question

Obtain the homogeneous wave equation in terms of the electric flux density D. Show your work please.

The answer is:

Answer 1

General Maxwell's equations rightarrow Nabla . E = 0 rightarrow Nabla times E = -partial differential/partial differential t B rightarrow Nabla times B = mu_0 E_) partial differential E/partial differential t rightarrow Nabla times (Nabla times E) = -partial differential/partial differential t Nabla times B = -mu_0 E_0 partial differential^2/partial differential t^2 (E) rightarrow Nabla times (Nabla times A) = Nabla (Nabla middot A) - Nabla^2 A Vector identities rightarrow Nabla^2 A = Nabla middot (Nabla A) rightarrow - Nabla^2 E = -mu_0 E_0 partial differential^2/partial differential t^2 E rightarrow Nabla^2 E = mu_0 E_0 partial differential^2/partial differential t^2 E rightarrow D = E rightarrow Replace E with Electric flux density rightarrow Nabla^2 D = mu_0 E partial differential^2/partial differential t^2 D rightarrow Nabla^2 D - muE partial differential^@/partial differential t^2 D = 0