choclo.prism.magnetic_n
choclo.prism.magnetic_n#
- choclo.prism.magnetic_n(easting, northing, upward, prism, magnetization)[source]#
Northing component of the magnetic field due to a prism
Returns the northing 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 (1d-array) – One dimensional array containing the coordinates of the prism in the following order:
west,east,south,north,bottom,topin a Cartesian coordinate system. All coordinates should be in meters.magnetization (1d-array) – Magnetization vector of the prism. It should have three components in the following order:
magnetization_easting,magnetization_northing,magnetization_upward. Should be in \(A m^{-1}\).
- Returns
b_n (float) – Northing component of the magnetic field generated by the prism on the observation point in \(\text{T}\).
Notes
Computes the northing 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:
\[B_y(\mathbf{p}) = \frac{\mu_0}{4\pi} \left( M_x u_{xy} + M_y u_{yy} + M_z u_{yz} \right)\]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]:
\[\begin{split}u_{xy} &= \Bigg\lvert\Bigg\lvert\Bigg\lvert \ln (z + r) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2} \\ u_{yy} &= \Bigg\lvert\Bigg\lvert\Bigg\lvert - \arctan \left( \frac{xz}{yr} \right) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2} \\ u_{yz} &= \Bigg\lvert\Bigg\lvert\Bigg\lvert \ln (x + r) \Bigg\rvert_{X_1}^{X_2} \Bigg\rvert_{Y_1}^{Y_2} \Bigg\rvert_{Z_1}^{Z_2}\end{split}\]References