choclo.prism.magnetic_e

choclo.prism.magnetic_e(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, magnetization_east, magnetization_north, magnetization_up)

Easting component of the magnetic field due to a prism

Returns the easting 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_e (float) – Easting 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.

Notes

Computes the easting 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:

$$B_x(\mathbf{p})=\frac{{\mu }_0}{4\pi }(M_x{u}_{xx}+M_y{u}_{xy}+M_z{u}_{xz})$$

Where $u_{ij}$ are:

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

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

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

References

• [Blakely1995]

• [Oliveira2015]

• [Nagy2000]

• [Nagy2002]

• [Fukushima2020]

See also

choclo.prism.magnetic_field, choclo.prism.magnetic_n, choclo.prism.magnetic_u