2.3 Magnetic field

The governing equation describing the magnetical part of magnetomechanical systems can be derived from Maxwell's equations. Due to the always solenoidal magnetic field, the magnetic flux density $\vec{B}$ can be expressed as the curl of the magnetic vector potential $\vec{A}$

 

$$\vec{B} = \nabla\times\vec{A}.$$ (5)

In the case of low frequencies, i.e. neglecting displacement current, magnetic fields can be described by the partial differential equation

 

$$\nabla\times\left(\frac{1}{\mu}\nabla\times\vec{A}\right) = \vec{J}_\mathrm{e}-\gamma \frac{\partial\vec{A}}{\partial t}-\gamma\nabla V,$$ (6)

where $\vec{J}_\mathrm{e}$ denotes the free current density, $\mu$ the permeability, V the scalar electric potential and $\gamma$ the electrical conductivity. The second term of the right-hand side of equation (6) represents the induced eddy current density in a resting electrically conductive body placed in a time-varying magnetic field. The third term of equation (6) expresses the current density due to the potential difference in a conductor.

In order to obtain a full description of the dynamic behavior of an electrodynamic loudspeaker, all coupling terms between the three physical fields have to be considered (see section 2.4 and 2.5).