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:
eastingfloat

Easting coordinate of the observation point. Must be in meters.

northingfloat

Northing coordinate of the observation point. Must be in meters.

upwardfloat

Upward coordinate of the observation point. Must be in meters.

prism_westfloat

The West boundary of the prism. Must be in meters.

prism_eastfloat

The East boundary of the prism. Must be in meters.

prism_southfloat

The South boundary of the prism. Must be in meters.

prism_northfloat

The North boundary of the prism. Must be in meters.

prism_bottomfloat

The bottom boundary of the prism. Must be in meters.

prism_topfloat

The top boundary 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_efloat

Easting 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.

b_nfloat

Northing 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.

b_ufloat

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.

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:

$\mathbf{B}(\mathbf{p}) = - \frac{\mu_0}{4\pi} \nabla_\mathbf{p} \left[ \int\limits_R \mathbf{M} \cdot \nabla_\mathbf{q} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} \right) dv \right]$

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

$B_x(\mathbf{p}) = - \frac{\mu_0}{4\pi} \left[ M_x \int\limits_R \frac{\partial}{\partial x_p} \left[ \frac{\partial}{\partial x_q} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} \right) \right] dv + M_y \int\limits_R \frac{\partial}{\partial x_p} \left[ \frac{\partial}{\partial y_q} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} \right) \right] dv + M_z \int\limits_R \frac{\partial}{\partial x_p} \left[ \frac{\partial}{\partial z_q} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} \right) \right] dv \right]$

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

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

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

$B_x(\mathbf{p}) = + \frac{\mu_0}{4\pi} \left[ M_x \frac{\partial^2}{\partial x_p^2} \int\limits_R \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} dv + M_y \frac{\partial^2}{\partial x_p \partial y_p} \int\limits_R \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} dv + M_z \frac{\partial^2}{\partial x_p \partial z_p} \int\limits_R \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} dv \right]$

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:

$u_{ij} = \frac{\partial}{\partial i} \frac{\partial}{\partial j} \int\limits_R \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert} dv$

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

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