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 (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_u (float) – Upward component of the magnetic field generated by the prism on the observation point in \(\text{T}\). Return
numpy.nanif 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