choclo.prism.magnetic_u#

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

Upward component of the magnetic field due to a prism.

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

Parameters:

easting, northing, upwardfloat

Easting, northing and upward coordinates of the observation point. Must be in meters.

prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_topfloat

The boundaries of the prism. Must be in meters.

magnetization_eastfloat

The East component of the magnetization vector of the prism. Must be in $A{m}^{-1}$.

magnetization_northfloat

The North component of the magnetization vector of the prism. Must be in $A{m}^{-1}$.

magnetization_upfloat

The upward component of the magnetization vector of the prism. Must be in $A{m}^{-1}$.

Returns:

b_ufloat

Upward component of the magnetic field generated by the prism on the observation point in $\text{T}$. Return numpy.nan if the observation point falls in a singular point: prism vertices, prism edges or interior points.

See also

choclo.prism.magnetic_field

choclo.prism.magnetic_e

choclo.prism.magnetic_n

choclo.prism.magnetic_ee

choclo.prism.magnetic_nn

choclo.prism.magnetic_uu

choclo.prism.magnetic_en

choclo.prism.magnetic_eu

choclo.prism.magnetic_nu

Notes

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

$$B_z(\mathbf{p})=\frac{{\mu }_0}{4\pi }(M_x{u}_{xz}+M_y{u}_{yz}+M_z{u}_{zz})$$

Where $u_{ij}$ are:

$$u_{ij}=\frac{{\partial }^2}{\partial i\partial j}\underset{R}{\int }\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }dv$$

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

$$\begin{array}{rl}{u}_{xz}& =|||\ln (y+r){|}_{X_1}^{X_2}{|}_{Y_1}^{Y_2}{|}_{Z_1}^{Z_2}\\ u_{yz}& =|||\ln (x+r){|}_{X_1}^{X_2}{|}_{Y_1}^{Y_2}{|}_{Z_1}^{Z_2}\\ u_{zz}& =|||-\arctan \left(\frac{xy}{zr}\right){|}_{X_1}^{X_2}{|}_{Y_1}^{Y_2}{|}_{Z_1}^{Z_2}\end{array}$$

References

• [Blakely1995]

• [Oliveira2015]

• [Nagy2000]

• [Nagy2002]

• [Fukushima2020]