# Node MagneticForceNode

Defined in `mime.nodes.magnetic._force.MagneticForceNode`.

Algorithm `MIME-NODE-107`.

The force of a field's gradient on a body's permanent moment, F = G^T m, and its torque about the centre of mass where the magnet sits off it.

## Governing equations

$$
\mathbf{F} = \mathbf{G}^{\top} \mathbf{m}, \quad \boldsymbol{\tau} = \mathbf{l} \times \mathbf{F}, \quad \mathbf{m} = R(\mathbf{q}) \, \mathbf{m}_{b}, \quad \mathbf{l} = R(\mathbf{q}) \, \mathbf{c}_{b}
$$

## Discretisation

none declared; a node whose order depends on its construction declares it with the `discretization_order()` hook, which is read from a built node.

## Assumptions

- A point dipole in a current-free field, the gradient read at the body's centre of mass
- A rigid, permanent moment

## Limitations

none declared

## Declared regimes (not evidenced here)

none declared

## References

[@Jackson1998]

## Symbols

| Symbol | Kind | Name | Units | Implemented by |
|---|---|---|---|---|
| $\mathbf{F}$ | state | `wrench` | N; N m | `mime.nodes.magnetic._force.MagneticForceNode.advance` |
| $\boldsymbol{\tau}$ | state | `wrench` | N; N m | `mime.nodes.magnetic._force.MagneticForceNode.advance` |
| $\mathbf{G}$ | input | `gradient` | T/m | `mime.nodes.magnetic._force.MagneticForceNode.advance` |
| $\mathbf{m}_{b}$ | parameter | `moment` | A m^2 | `mime.nodes.magnetic._force.MagneticForceNode.advance` |
| $\mathbf{q}$ | input | `pose` | m; 1 | `mime.nodes.magnetic._force.MagneticForceNode.advance` |
| $\mathbf{c}_{b}$ | parameter | `magnet_centre` | m | `mime.nodes.magnetic._force.MagneticForceNode.advance` |

## Built by

- [`magnetic.force`](../entries/interaction/magnetic.force.md) 0.1.0