5.1.3 Quasi-Magnetostatics

If the characteristic lengths of the analyzed structures are much smaller than the considered wave lengths and for conducting areas $\epsilon / \gamma \ll T$ , the displacement current can be neglected [40,69]. $T$ is the characteristic period of the time change rate. In this case the Maxwell equations (4.1) to (4.4) can be simplified and the so called dominant magnetic field model is achieved:

$$\vec{\nabla}\times\vec{E} = -\mu\partial_t\vec{H}$$ (5.11)

$$\vec{\nabla}\cdot\vec{B} = 0$$ (5.12)

$$\vec{\nabla}\times\vec{H} = \vec{J}.$$ (5.13)

After transforming (5.13)

$$\vec{\nabla}\times\vec{H} = \vec{J} = \gamma\vec{E} \Rightarrow \frac{1}{\gamma}\vec{\nabla}\times\vec{H} = \vec{E}$$ (5.14)

the rotor operator is applied and the right hand side is substituted by (5.11)

$$\vec{\nabla}\times\left(\frac{1}{\gamma}\vec{\nabla}\times\vec{H}\right) = \vec{\nabla}\times\vec{E} = -\partial_t\vec{B} = -\mu\partial_t\vec{H}.$$ (5.15)

Thus, in the frequency domain the quasi-magnetostatic case is described by the following partial differential equation for the magnetic field $\vec{H}$

$$\vec{\nabla}\times\left(\frac{1}{\gamma}\vec{\nabla}\times\vec{H}\right) + \jmath\omega\mu\vec{H} = \vec{0}.$$ (5.16)