Electrical Dipole Gradient Visualization with Matplotlib

Introduction

In this lab, you will learn how to use Matplotlib to create a gradient plot of an electrical dipole. You will learn how to create a triangulation, refine data, and compute the electrical field. Finally, you will plot the triangulation, the potential iso-contours, and the vector field.

VM Tips

After the VM startup is done, click the top left corner to switch to the Notebook tab to access Jupyter Notebook for practice.

Sometimes, you may need to wait a few seconds for Jupyter Notebook to finish loading. The validation of operations cannot be automated because of limitations in Jupyter Notebook.

If you face issues during learning, feel free to ask Labby. Provide feedback after the session, and we will promptly resolve the problem for you.

Create the x and y coordinates of the points

n_angles = 30
n_radii = 10
min_radius = 0.2
radii = np.linspace(min_radius, 0.95, n_radii)

angles = np.linspace(0, 2 * np.pi, n_angles, endpoint=False)
angles = np.repeat(angles[..., np.newaxis], n_radii, axis=1)
angles[:, 1::2] += np.pi / n_angles

x = (radii*np.cos(angles)).flatten()
y = (radii*np.sin(angles)).flatten()

Explanation:

• n_angles is the number of angles in a circle.
• n_radii is the number of circles.
• min_radius is the minimum radius of the circles.
• radii is an array of radii.
• angles is an array of angles.
• x is an array of x-coordinates.
• y is an array of y-coordinates.

Compute the electrical potential of a dipole

def dipole_potential(x, y):
    """The electric dipole potential V, at position *x*, *y*."""
    r_sq = x**2 + y**2
    theta = np.arctan2(y, x)
    z = np.cos(theta)/r_sq
    return (np.max(z) - z) / (np.max(z) - np.min(z))

V = dipole_potential(x, y)

Explanation:

• dipole_potential is a function that calculates the electrical dipole potential.
• V is an array of electrical dipole potentials.

Create the Triangulation

triang = Triangulation(x, y)

triang.set_mask(np.hypot(x[triang.triangles].mean(axis=1),
                         y[triang.triangles].mean(axis=1))
                < min_radius)

Explanation:

• Triangulation is a class that creates a Delaunay triangulation from a set of points.
• triang is an instance of the Triangulation class.
• triang.set_mask masks off unwanted triangles.

Refine data

refiner = UniformTriRefiner(triang)
tri_refi, z_test_refi = refiner.refine_field(V, subdiv=3)

Explanation:

• UniformTriRefiner is a class that refines a triangulation to create a more accurate plot.
• refiner is an instance of the UniformTriRefiner class.
• tri_refi and z_test_refi are refined triangulation and potential values, respectively.

Compute the electrical field

tci = CubicTriInterpolator(triang, -V)

(Ex, Ey) = tci.gradient(triang.x, triang.y)
E_norm = np.sqrt(Ex**2 + Ey**2)

Explanation:

• CubicTriInterpolator is a class that interpolates data using a cubic polynomial.
• tci is an instance of the CubicTriInterpolator class.
• (Ex, Ey) is the electrical field.
• E_norm is the normalized electrical field.

Plot the triangulation, the potential iso-contours, and the vector field

fig, ax = plt.subplots()
ax.set_aspect('equal')
ax.use_sticky_edges = False
ax.margins(0.07)

ax.triplot(triang, color='0.8')

levels = np.arange(0., 1., 0.01)
ax.tricontour(tri_refi, z_test_refi, levels=levels, cmap='hot',
              linewidths=[2.0, 1.0, 1.0, 1.0])

ax.quiver(triang.x, triang.y, Ex/E_norm, Ey/E_norm,
          units='xy', scale=10., zorder=3, color='blue',
          width=0.007, headwidth=3., headlength=4.)

ax.set_title('Gradient Plot: Electrical Dipole')
plt.show()

Explanation:

• fig and ax are the figure and axes objects, respectively.
• ax.set_aspect sets the aspect ratio of the axes.
• ax.use_sticky_edges and ax.margins set the margins of the axes.
• ax.triplot plots the triangulation.
• ax.tricontour plots the potential iso-contours.
• ax.quiver plots the vector field.
• ax.set_title sets the title of the plot.

Summary

In this lab, you learned how to use Matplotlib to create a gradient plot of an electrical dipole. You learned how to create a triangulation, refine data, and compute the electrical field. Finally, you plotted the triangulation, the potential iso-contours, and the vector field.