which fulfills eqn. (2.2) since evaluates to zero for every vector field . Inserting eqn. (2.7) into eqn. (2.1) gives

(2.8) |

Interchanging the order of the time derivative and the curl operator,

(2.9) |

and using the associative property of the curl operator,

(2.10) |

the argument of the curl operator can be substituted by the gradient of a scalar potential

since yields zero for every scalar field . The minus sign on the right hand side of eqn. (2.11) is introduced by convention based on historical reasons.

In the quasi-stationary case, which holds true for semiconductor devices^{2.1}, the time
derivative of the vector potential can be neglected

POISSON's equation is finally obtained by inserting eqn. (2.12) into eqn. (2.5)

(2.13) |

which is in turn inserted into eqn. (2.4)

In the case of vanishing space charge density POISSON's equation simplifies to the LAPLACE equation

(2.15) |