choclo.prism.magnetic_field#

choclo.prism.magnetic_field(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, magnetization_east, magnetization_north, magnetization_up)[source]#

Magnetic field due to a rectangular prism.

Returns the three components of the magnetic field due to a single rectangular prism on a single computation point.

Note

Use this function when all the three component of the magnetic fields are needed. Running this function is faster than computing each component separately. Use one of magnetic_e, magnetic_n, magnetic_u if you need only one of them.

Parameters:

easting, northing, upwardfloat

Easting, northing and upward coordinates of the observation point. Must be in meters.

prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_topfloat

The boundaries 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_e, b_n, b_ufloat

Easting, northing and upward component of the magnetic field generated by the prism on the observation point in $\text{T}$. It will be numpy.nan if the observation point falls in a singular point: prism vertices, prism edges or interior points.

See also

choclo.prism.magnetic_e

choclo.prism.magnetic_n

choclo.prism.magnetic_u

choclo.prism.magnetic_ee

choclo.prism.magnetic_nn

choclo.prism.magnetic_uu

choclo.prism.magnetic_en

choclo.prism.magnetic_eu

choclo.prism.magnetic_nu

Notes

Consider an observation point $\mathbf{p}$ and a prism $R$ with a magnetization vector $\mathbf{M}$. The magnetic field $\mathbf{B}$ it generates on the observation point $\mathbf{p}$ is defined as:

$$\mathbf{B}(\mathbf{p})=-\frac{{\mu }_0}{4\pi }{\mathrm{\nabla }}_{\mathbf{p}}[\underset{R}{\int }\mathbf{M}\cdot {\mathrm{\nabla }}_{\mathbf{q}}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }\right)dv]$$

Since the magnetization vector is constant inside the boundaries of the prism, we can write the easting component of $\mathbf{B}$ as:

$$B_x(\mathbf{p})=-\frac{{\mu }_0}{4\pi }[M_x\underset{R}{\int }\frac{\partial }{\partial x_p}\left[\frac{\partial }{\partial x_q}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }\right)\right]dv+M_y\underset{R}{\int }\frac{\partial }{\partial x_p}\left[\frac{\partial }{\partial y_q}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }\right)\right]dv+M_z\underset{R}{\int }\frac{\partial }{\partial x_p}\left[\frac{\partial }{\partial z_q}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }\right)\right]dv]$$

where $M_x$, $M_y$ and $M_z$ are the components of the magnetization vector. The other components can be expressed in an analogous way.

It can be proved that

$$\frac{\partial }{\partial x_q}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }\right)=-\frac{\partial }{\partial x_p}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }\right)$$

and that it also holds for the two other directions. Therefore, we can rewrite $B_x$ as:

$$B_x(\mathbf{p})=+\frac{{\mu }_0}{4\pi }[M_x\frac{{\partial }^2}{\partial x_p^2}\underset{R}{\int }\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }dv+M_y\frac{{\partial }^2}{\partial x_p\partial y_p}\underset{R}{\int }\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }dv+M_z\frac{{\partial }^2}{\partial x_p\partial z_p}\underset{R}{\int }\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }dv]$$

Solutions to each one of the integrals in the previous equation and their second derivatives are given by [Nagy2000].

Following [Oliveira2015] we can define a symmetrical 3x3 matrix $\mathbf{U}$ whose elements are the second derivatives of the previous integrals, such as:

$$u_{ij}=\frac{{\partial }^2}{\partial i\partial j}\underset{R}{\int }\frac{1}{\Vert \mathbf{p}-\mathbf{q}\Vert }dv$$

with $i,j\in \{x,y,z\}$.

We can then express the magnetic field $\mathbf{B}(\mathbf{p})$ generated by the prism in a compact form:

$$\mathbf{B}(\mathbf{p})=\frac{{\mu }_0}{4\pi }\mathbf{U}\cdot \mathbf{M}$$

References

• [Blakely1995]

• [Oliveira2015]

• [Nagy2000]

• [Nagy2002]

• [Fukushima2020]