choclo.prism.magnetic_n#

choclo.prism.magnetic_n(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, magnetization_east, magnetization_north, magnetization_up)[source]#

Northing component of the magnetic field due to a prism

Returns the northing component of the magnetic field due to a single rectangular prism on a single computation point.

Parameters:

easting (float) – Easting coordinate of the observation point. Must be in meters.
northing (float) – Northing coordinate of the observation point. Must be in meters.
upward (float) – Upward coordinate of the observation point. Must be in meters.
prism_west (float) – The West boundary of the prism. Must be in meters.
prism_east (float) – The East boundary of the prism. Must be in meters.
prism_south (float) – The South boundary of the prism. Must be in meters.
prism_north (float) – The North boundary of the prism. Must be in meters.
prism_bottom (float) – The bottom boundary of the prism. Must be in meters.
prism_top (float) – The top boundary of the prism. Must be in meters.
magnetization_east (float) – The East component of the magnetization vector of the prism. Must be in $A m^{- 1}$ .
magnetization_north (float) – The North component of the magnetization vector of the prism. Must be in $A m^{- 1}$ .
magnetization_up (float) – The upward component of the magnetization vector of the prism. Must be in $A m^{- 1}$ .

Returns:

b_n (float) – Northing component of the magnetic field generated by the prism on the observation point in $T$ . Return numpy.nan if the observation point falls in a singular point: prism vertices, prism edges or interior points.

Notes

Computes the northing component of the magnetic field $B (p)$ generated by a rectangular prism $R$ with a magnetization vector $M$ on the observation point $p$ as follows:

B_{y} (p) = \frac{μ_{0}}{4 π} (M_{x} u_{x y} + M_{y} u_{y y} + M_{z} u_{y z})

Where $u_{i j}$ are:

u_{i j} = \frac{\partial}{\partial i} \frac{\partial}{\partial j} \int_{R} \frac{1}{‖ p - q ‖} d v

with $i, j \in {x, y, z}$ . Solutions of the second derivatives of these integrals are given by [Nagy2000]:

\begin{aligned} u_{x y} & = | | | \ln (z + r) |_{X_{1}}^{X_{2}} |_{Y_{1}}^{Y_{2}} |_{Z_{1}}^{Z_{2}} \\ u_{y y} & = | | | - \arctan (\frac{x z}{y r}) |_{X_{1}}^{X_{2}} |_{Y_{1}}^{Y_{2}} |_{Z_{1}}^{Z_{2}} \\ u_{y z} & = | | | \ln (x + r) |_{X_{1}}^{X_{2}} |_{Y_{1}}^{Y_{2}} |_{Z_{1}}^{Z_{2}} \end{aligned}

References

choclo.prism.magnetic_n

Contents

choclo.prism.magnetic_n#