4.1.1 Maxwell's Equations

The basic equations solved in a device simulator can be derived from Maxwell's equations. The four partial differential equations relate the electric field ( $\ensuremath{\mathitbf{E}}$), the displacement field ( $\ensuremath{\mathitbf{D}}$), the magnetic field ( $\ensuremath{\mathitbf{H}}$), and the induction field ( $\ensuremath{\mathitbf{B}}$) vectors to the current density ( $\ensuremath{\mathitbf{J}}$) and the electric charge density ( $\ensuremath{\rho}$):

$$\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\times} \ensuremath{\mathitbf{H}} = \ensuremath{\mathitbf{J}} + \frac{\partial \ensuremath{\mathitbf{D}}}{\partial t}$$ (4.1)

$$\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\times} \ensuremath{\mathitbf{E}} = - \frac{\partial \ensuremath{\mathitbf{B}}}{\partial t}$$ (4.2)

$$\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremath{\cdot}} \ensuremath{\mathitbf{D}} = \ensuremath{\rho}$$ (4.3)

$$\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla}}}\ensuremath{\cdot}} \ensuremath{\mathitbf{B}} = 0.$$ (4.4)