# geoana.em.tdem.vertical_magnetic_field_time_deriv_horizontal_loop#

geoana.em.tdem.vertical_magnetic_field_time_deriv_horizontal_loop(t, sigma=1.0, mu=1.25663706212e-06, radius=1.0, current=1.0, turns=1)#

Time-derivative of the vertical transient magnetic field at the center of a horizontal loop over a halfspace.

Compute the time-derivative of the vertical component of the transient magnetic field at the center of a circular loop on the surface of a conductive and magnetically permeable halfspace.

Parameters:
tfloat, or numpy.ndarray
sigmafloat, optional

conductivity

mufloat, optional

magnetic permeability

currentfloat, optional

current of the horizontal loop

turnsint, optional

number of turns in the horizontal loop

Returns:
dhz_dtfloat, or numpy.ndarray

The vertical magnetic field time derivative at the center of the loop. The shape will match the t input.

Notes

Matches equation 4.97 of Ward and Hohmann 1988.

$\frac{\partial h_z}{\partial t} = -\frac{I}{\sigma a^3}\left[ 3 \mathrm{erf}(\theta a) - \frac{2}{\sqrt{\pi}}\theta a (3 + 2 \theta^2 a^2)e^{-\theta^2 a^2} \right]$

Examples

Reproducing part of Figure 4.8 from Ward and Hohmann 1988

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from geoana.em.tdem import vertical_magnetic_field_time_deriv_horizontal_loop


Calculate the field at the time given

>>> times = np.logspace(-7, -1)
>>> dhz_dt = vertical_magnetic_field_time_deriv_horizontal_loop(times, sigma=1E-2, radius=50)


Match the vertical magnetic field plot

>>> plt.loglog(times*1E3, -dhz_dt, '--')
>>> plt.xlabel('time (ms)')
>>> plt.ylabel(r'$\frac{\partial h_z}{ \partial t}$ (A/(m s)')
>>> plt.show()