# Magnetic field lines example

This example uses the streamline module to display field lines of a magnetic dipole (a current loop).

This example requires scipy.

The magnetic field from an arbitrary current loop is calculated from eqns (1) and (2) in Phys Rev A Vol. 35, N 4, pp. 1535-1546; 1987.

To get a prettier result, we use a fairly large grid to sample the field. As a consequence, we need to clear temporary arrays as soon as possible.

For a more thorough example of magnetic field calculation and visualization with Mayavi and scipy, see Magnetic field example.

Python source code: `magnetic_field_lines.py`

    # Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
    # Copyright (c) 2007, Enthought, Inc.
    # License: BSD Style.

    import numpy as np
    from scipy import special

    #### Calculate the field ####################################################
    radius = 1  # Radius of the coils

    x, y, z = [e.astype(np.float32) for e in
                np.ogrid[-10:10:150j, -10:10:150j, -10:10:150j]]

    # express the coordinates in polar form
    rho = np.sqrt(x ** 2 + y ** 2)
    x_proj = x / rho
    y_proj = y / rho
    # Free memory early
    del x, y

    E = special.ellipe((4 * radius * rho) / ((radius + rho) ** 2 + z ** 2))
    K = special.ellipk((4 * radius * rho) / ((radius + rho) ** 2 + z ** 2))
    Bz = 1 / np.sqrt((radius + rho) ** 2 + z ** 2) * (
                    K
                    + E * (radius ** 2 - rho ** 2 - z ** 2) /
                        ((radius - rho) ** 2 + z ** 2)
                )
    Brho = z / (rho * np.sqrt((radius + rho) ** 2 + z ** 2)) * (
                    - K
                    + E * (radius ** 2 + rho ** 2 + z ** 2) /
                        ((radius - rho) ** 2 + z ** 2)
                )
    del E, K, z, rho
    # On the axis of the coil we get a divided by zero. This returns a
    # NaN, where the field is actually zero :
    Brho[np.isnan(Brho)] = 0

    Bx, By = x_proj * Brho, y_proj * Brho

    del x_proj, y_proj, Brho

    #### Visualize the field ####################################################
    from mayavi import mlab
    fig = mlab.figure(1, size=(400, 400), bgcolor=(1, 1, 1), fgcolor=(0, 0, 0))

    field = mlab.pipeline.vector_field(Bx, By, Bz)
    # Unfortunately, the above call makes a copy of the arrays, so we delete
    # this copy to free memory.
    del Bx, By, Bz

    magnitude = mlab.pipeline.extract_vector_norm(field)
    contours = mlab.pipeline.iso_surface(magnitude,
                                            contours=[0.01, 0.8, 3.8, ],
                                            transparent=True,
                                            opacity=0.4,
                                            colormap='YlGnBu',
                                            vmin=0, vmax=2)

    field_lines = mlab.pipeline.streamline(magnitude, seedtype='line',
                                            integration_direction='both',
                                            colormap='bone',
                                            vmin=0, vmax=1)

    # Tweak a bit the streamline.
    field_lines.stream_tracer.maximum_propagation = 100.
    field_lines.seed.widget.point1 = [69, 75.5, 75.5]
    field_lines.seed.widget.point2 = [82, 75.5, 75.5]
    field_lines.seed.widget.resolution = 50
    field_lines.seed.widget.enabled = False

    mlab.view(42, 73, 104, [79,  75,  76])

    mlab.show()