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, upward
float Easting, northing and upward coordinates of the observation point. Must be in meters.
- prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top
float The boundaries 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 \(A m^{-1}\).
- easting, northing, upward
- 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.
- b_u
See also
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:
\[u_{ij} = \frac{\partial^2}{\partial i \partial j} \int\limits_R \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} dv\]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