# choclo.prism.magnetic_u#

choclo.prism.magnetic_u(easting, northing, upward, prism, magnetization)[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 (1d-array) – One dimensional array containing the coordinates of the prism in the following order: west, east, south, north, bottom, top in 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_u (float) – Upward component of the magnetic field generated by the prism on the observation point in $$\text{T}$$.

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