PDE 28: Maxwell's equations

In one space dimension, the equations are \[E_t = c B_x, \quad B_t = c E_x. \] with \(x \in [0,1]\) and \(t > 0\), and where \(E = E(x,t)\) is the electric field and \(B = B(x,t)\) is the magnetic field. One could consider boundary conditions \(E(0,t) = E(1,t) = 0\) and \(B_x(0,t) = B_x(1,t) = 0.\)

In 3D, the equations are \[E_t = c \nabla \times B, \quad B_t = -c \nabla \times E, \] with \(E = E(x,y,z,t)\) and \(B = B(x,y,z,t)\). Here \( \times\) is the cross product of 3-vectors, and \(\nabla \times\) is the curl operator. These equations can be solved e.g. by considering periodic boundary conditions on the unit cube.

Google-åte: Maxwell's equations, electromagnetism.