choclo.prism.magnetic_u

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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:
eastingfloat

Easting coordinate of the observation point. Must be in meters.

northingfloat

Northing coordinate of the observation point. Must be in meters.

upwardfloat

Upward coordinate of the observation point. Must be in meters.

prism_westfloat

The West boundary of the prism. Must be in meters.

prism_eastfloat

The East boundary of the prism. Must be in meters.

prism_southfloat

The South boundary of the prism. Must be in meters.

prism_northfloat

The North boundary of the prism. Must be in meters.

prism_bottomfloat

The bottom boundary of the prism. Must be in meters.

prism_topfloat

The top boundary 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.

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:

\[B_z(\mathbf{p}) = \frac{\mu_0}{4\pi} \left( M_x u_{xz} + M_y u_{yz} + M_z u_{zz} \right)\]

Where \(u_{ij}\) are:

\[u_{ij} = \frac{\partial}{\partial i} \frac{\partial}{\partial j} \int\limits_R \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} dv\]

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

\[\begin{split}u_{xz} &= \Bigg\lvert\Bigg\lvert\Bigg\lvert \ln (y + r) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2} \\ u_{yz} &= \Bigg\lvert\Bigg\lvert\Bigg\lvert \ln (x + r) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2} \\ u_{zz} &= \Bigg\lvert\Bigg\lvert\Bigg\lvert - \arctan \left( \frac{xy}{zr} \right) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2}\end{split}\]

References