geoana.em.static.DipoleHalfSpace.electric_field#

DipoleHalfSpace.electric_field(xyz_m, xyz_n=None)#

Electric field for a dipole source in a halfspace.

This method computes the electric field for a dipole source in a halfspace at the set of gridded xyz locations provided. Where \(-\nabla V\) is the negative gradient of the electric potential for a dipole source. The electric field \(\mathbf{E}\) is:

\[\mathbf{E} = -\nabla V\]
Parameters
xyz_m(…, 3) numpy.ndarray

Location of the M voltage electrode.

xyz_n(…, 3) numpy.ndarray, optional

Location of the N voltage electrode.

Returns
E(…, 3) np.ndarray

Electric field of point current in units \(\frac{V}{m}\).

Examples

Here, we define a dipole source in a halfspace to compute electric field.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from mpl_toolkits.axes_grid1 import make_axes_locatable
>>> from geoana.em.static import DipoleHalfSpace

Define the dipole source.

>>> rho = 1.0
>>> current = 1.0
>>> location_a = np.r_[-1, 0, 0]
>>> location_b = np.r_[1, 0, 0]
>>> simulation = DipoleHalfSpace(
>>>     current=current, rho=rho, location_a=location_a, location_b=location_b
>>> )

Now we create a set of gridded locations and compute the electric field.

>>> X, Y = np.meshgrid(np.linspace(-2, 2, 20), np.linspace(-2, 2, 20))
>>> Z = np.zeros_like(X)
>>> xyz = np.stack((X, Y, Z), axis=-1)
>>> e1 = simulation.electric_field(xyz)
>>> e2 = simulation.electric_field(xyz - np.r_[2, 0, 0], xyz + np.r_[2, 0, 0])

Finally, we plot the electric field.

>>> fig, axs = plt.subplots(1, 2, figsize=(18,12))
>>> titles = ['3 Electrodes', '4 Electrodes']
>>> for ax, E, title in zip(axs.flatten(), [e1, e2], titles):
>>>     E_amp = np.linalg.norm(E, axis=-1)
>>>     im = ax.pcolor(X, Y, E_amp, shading='auto')
>>>     divider = make_axes_locatable(ax)
>>>     cax = divider.append_axes("right", size="5%", pad=0.05)
>>>     cb = plt.colorbar(im, cax=cax)
>>>     cb.set_label(label= 'Amplitude ($V/m$)')
>>>     ax.streamplot(X, Y, E[..., 0], E[..., 1], density=0.75)
>>>     ax.set_ylabel('Y coordinate ($m$)')
>>>     ax.set_xlabel('X coordinate ($m$)')
>>>     ax.set_aspect('equal')
>>>     ax.set_title(title)

Finally, we plot the electric field.

>>> E_amp = np.linalg.norm(e1, axis=-1)
>>> plt.pcolor(X, Y, E_amp, shading='auto')
>>> cb = plt.colorbar()
>>> cb.set_label(label= 'Electric Field ($V/m$)')
>>> plt.streamplot(X, Y, e1[..., 0], e1[..., 1], density=0.75)
>>> plt.ylabel('Y coordinate ($m$)')
>>> plt.xlabel('X coordinate ($m$)')
>>> plt.title('Electric Field from Dipole using 3 Electrodes')
>>> plt.tight_layout()
>>> plt.show()

(Source code, png, pdf)

../../_images/geoana-em-static-DipoleHalfSpace-electric_field-1.png