Problem 4: Every solution of Maxwell's equations in free space solves the wave equa- tion (both E and B do). The converse is not true. However, in this (and the next) exercise we show that if E and B are solutions to the wave equation and in addition, they satisfy Maxwell's equations at time t = 0, then they are solutions to Maxwell's equations. Assume that E, B : R³ × R → R³, E, B € C³(R³), are solutions to the wave equation - c² Au= 0. J²u Ət² JE JE 1. Show that , i = 1, 2, 3, and are solutions to the wave equation and conclude ?х; that the corresponding claim holds true for B. 2. Show that V. E, V.B, equation. JE - OB c²V × B and + V × E are solutions to the wave Ət

Problem 4: Every solution of Maxwell's equations in free space solves the wave equa- tion (both E and B do). The converse is not true. However, in this (and the next) exercise we show that if E and B are solutions to the wave equation and in addition, they satisfy Maxwell's equations at time t = 0, then they are solutions to Maxwell's equations. Assume that E, B : R³ × R → R³, E, B € C³(R³), are solutions to the wave equation - c² Au= 0. J²u Ət² JE JE 1. Show that , i = 1, 2, 3, and are solutions to the wave equation and conclude ?х; that the corresponding claim holds true for B. 2. Show that V. E, V.B, equation. JE - OB c²V × B and + V × E are solutions to the wave Ət