# Kratos Magnetostatic Application

## General Description

The KRATOS Magnetostatic Application deals with the Finite Element Analysis of problems in magnetostatics. Though the mathematics and general formulation is quite similar to Electrostatics, Magnetostatics presents important differences at a conceptual and a programming level. Therefore, if you are a beginner in electromagnetism or programming electromagnetic applications, we strongly recommend you that you first start with the Electrostatic application.

The following description of the Kratos Magnetostatic Application takes part of:

At a first glance, it could be said that Electrostatics works with static charges whereas Magnetostatics does it with dynamic charges (or static currents). That's partially true and we avoid by now to

, but we are going to consider Magnetostatics as a completely new phenomena instead of thinking in a kind of electrodynamics.

Magnetostatics

Electrostatic establishes the most basic principles of the Maxwell equations and it is excellent to work at academic level. Nevertheless, some serious industrial applications can be solved by using just these simplest equations and simulation program (circuit components such as condensers and bipolar diodes, electrical field distribution for high voltage systems, etc).

## Theory

Magnetostatics refers to the physical phenomena related to the presence of electric charges in the objects.

### Basic principles Our first Maxwell's equation: $\vec{\nabla}\cdot\vec{D} = \rho$

where: $\vec{D} = \varepsilon\vec{E} = \varepsilon_0\varepsilon_r\vec{E}$

with: $\varepsilon, \varepsilon_0, \varepsilon_r$ the permittivity of the medium, of the free space and relative permittivity, respectively.

The electric field is the gradient of an electrostatic potential ( $V [Volts]\,$): $\vec{E}=-\vec{\nabla} V$

Our second Maxwell's equation: $\vec{\nabla}\times\vec{E} = 0$

By combining these Maxwell's equations, we can easily can obtain the Poisson's equation: $\vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0$

Basic principles extended

### Poisson's Equation in Magnetostatics

The detailed form of the Poisson's Equation in Magnetostatics is: $\frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial V(x,y,z)}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial V(x,y,z)}{\partial y} \right) + \frac{\partial}{\partial z}\cdot \left(\varepsilon_{z} \cdot \frac{\partial V(x,y,z)}{\partial z} \right) + \rho_v(x,y,z)=0$

Poisson's Equation extended

### Magnetostatic Boundary Conditions

For a domain ${\Omega} \,$, we should consider three different boundary conditions:

• Dirichlet boundary condition: $\left . V - \bar V = 0 \right |_{\Gamma_{V}}$
• Neumann boundary condition: $\left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}}$
• Infinit condition (when no physical boundary are presents -free space-): $\left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r}$

Boundary Conditions extended