Question

9. (a)Using the Maxwell's equations prove that the wave equations for the electric and magnetic fields are given by 0t2 where l/c-μοεο

Answer 1

Maxwell's equation in free space are given by (1) nabla. E^vector = 0 (Gauss law) (2) nabla times E^vector = -partial differential B^vector/partial differential t [Faraday's law] (3) nabla. B^vector = 0 magnetic monopoles (4) nabla times B^vector = mu_0 epsilon_0 partial differential E^vector/partial differential t (Ampere's law) * For electric field: - from Faraday law nabla times E^vector - partial differential B^vector/partial differential t (a) Taking curl on both sides to eqn (a) implies nabla times (nabla times E^vector) = nabla times (- partial differential B^vector/partial differential t) implies nabla (nabla. E^vector) - nabla^2 E^vector = -partial differential B^vector/partial differential t - (a) Taking curl on both sides to eqn (a) implies nabla times (nabla times E^vector) = nabla times (- partial differential B/partial differential t) implies nabla^2 E^vector = nabla times (- partial differential B^vector/partial differential t) implies - nabla^2 E^vector = -partial differential/partial differential t (nabla times B) [Because nabla times B^vector = mu_0 epsilon_0 partial differential E^vector/ partial differential t] implies - nabla^2 E^vector = -partial differential/partial differential