geoana.em.static.MagneticDipoleWholeSpace.vector_potential#

MagneticDipoleWholeSpace.vector_potential(xyz, coordinates='cartesian')#

Compute the vector potential for the static magnetic dipole.

This method computes the vector potential for the magnetic dipole at the set of gridded xyz locations provided. Where $$\mu$$ is the magnetic permeability, $$\mathbf{m}$$ is the dipole moment, $$\mathbf{r_0}$$ the dipole location and $$\mathbf{r}$$ is the location at which we want to evaluate the vector potential $$\mathbf{a}$$:

$\mathbf{a}(\mathbf{r}) = \frac{\mu}{4\pi} \frac{\mathbf{m} \times \, \Delta \mathbf{r}}{| \Delta r |^3}$

where

$\mathbf{\Delta r} = \mathbf{r} - \mathbf{r_0}$

For reference, see equation 5.83 in Griffiths (1999).

Parameters
xyz(n, 3) numpy.ndarray xyz

gridded locations at which we are calculating the vector potential

coordinates: str {‘cartesian’, ‘cylindrical’}

coordinate system that the location (xyz) are provided. The solution is also returned in this coordinate system. Default: “cartesian”

Returns
(n, 3) numpy.ndarray

The magnetic vector potential at each observation location in the coordinate system specified in units Tm.

Examples

Here, we define a z-oriented magnetic dipole and plot the vector potential on the xy-plane that intercepts at z=0.

>>> from geoana.em.static import MagneticDipoleWholeSpace
>>> from geoana.utils import ndgrid
>>> from geoana.plotting_utils import plot2Ddata
>>> import numpy as np
>>> import matplotlib.pyplot as plt


Let us begin by defining the magnetic dipole.

>>> location = np.r_[0., 0., 0.]
>>> orientation = np.r_[0., 0., 1.]
>>> moment = 1.
>>> dipole_object = MagneticDipoleWholeSpace(
>>>     location=location, orientation=orientation, moment=moment
>>> )


Now we create a set of gridded locations and compute the vector potential.

>>> xyz = ndgrid(np.linspace(-1, 1, 20), np.linspace(-1, 1, 20), np.array([0]))
>>> a = dipole_object.vector_potential(xyz)


Finally, we plot the vector potential on the plane. Given the symmetry, there are only horizontal components.

>>> fig = plt.figure(figsize=(4, 4))
>>> ax = fig.add_axes([0.15, 0.15, 0.8, 0.8])
>>> plot2Ddata(xyz[:, 0:2], a[:, 0:2], ax=ax, vec=True, scale='log')
>>> ax.set_xlabel('X')
>>> ax.set_ylabel('Z')
>>> ax.set_title('Vector potential at z=0')