.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/fields/electrodynamics/plot_dipole_radiation.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_fields_electrodynamics_plot_dipole_radiation.py: A Hertzian dipole antenna radiating on the Yee grid ======================================================== Twenty-two years after Maxwell's equations predicted electromagnetic radiation, Heinrich Hertz built an oscillating dipole antenna -- an induction coil driving a spark gap -- and detected the resulting waves meters away, confirming they travel at the speed of light and can be reflected, refracted, and polarized just like ordinary light. On the 2D Yee grid used here, the out-of-plane field :math:`E_z` and the in-plane fields :math:`H_x`, :math:`H_y` obey the source-free TMz-mode reduction of Maxwell's curl equations, .. math:: \frac{\partial H_x}{\partial t} = -\frac{1}{\mu_0}\frac{\partial E_z}{\partial y}, \qquad \frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\frac{\partial E_z}{\partial x}, \qquad \frac{\partial E_z}{\partial t} = \frac{1}{\varepsilon_0}\left(\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y}\right), and :func:`~physicskit.fields.electrodynamics.oscillating_dipole_source` builds a soft point source -- Hertz's spark-gap transmitter, in miniature -- that adds :math:`A\sin(2\pi f t)` directly into :math:`E_z` at a single grid cell every step, standing in for the driving antenna current. :func:`~physicskit.fields.electrodynamics.fdtd_2d_tmz_evolve` records the outgoing ``Ez`` field as it radiates away from the antenna on the Yee grid, frame by frame. .. GENERATED FROM PYTHON SOURCE LINES 28-34 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.fields import animate_field_2d, courant_limit_2d, fdtd_2d_tmz_evolve, oscillating_dipole_source .. GENERATED FROM PYTHON SOURCE LINES 35-38 An oscillating dipole source at the center of an empty 2D Yee grid -- Hertz's spark-gap transmitter, in miniature ------------------------------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 38-47 .. code-block:: Python Nx, Ny, dx, dy = 100, 100, 1e-3, 1e-3 dt = 0.5 * courant_limit_2d(dx, dy) Ez0 = np.zeros((Nx, Ny)) Hx0 = np.zeros((Nx, Ny)) Hy0 = np.zeros((Nx, Ny)) eps_r, mu_r = np.ones((Nx, Ny)), np.ones((Nx, Ny)) source = oscillating_dipole_source(Nx // 2, Ny // 2, amplitude=1.0, freq=3e10) .. GENERATED FROM PYTHON SOURCE LINES 48-50 Record snapshots as the field radiates outward from the antenna -------------------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 50-54 .. code-block:: Python steps = 140 frames, times = fdtd_2d_tmz_evolve(Ez0, Hx0, Hy0, eps_r, mu_r, steps=steps, dt=dt, dx=dx, dy=dy, source=source, snapshot_stride=2) .. GENERATED FROM PYTHON SOURCE LINES 55-58 Just as Hertz's receiver detected sparks far from the transmitter, energy from the dipole has visibly spread across the grid as an outgoing circular wavefront by the final frame. .. GENERATED FROM PYTHON SOURCE LINES 58-66 .. code-block:: Python x = np.arange(Nx) * dx y = np.arange(Ny) * dy X, Y = np.meshgrid(x, y, indexing="ij") anim = animate_field_2d(X, Y, frames, times) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 67-71 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("dipole_radiation.gif", writer="pillow", fps=20) .. GENERATED FROM PYTHON SOURCE LINES 71-76 .. code-block:: Python radius_cells = frames.shape[1] // 2 - 5 outer_ring_energy = np.abs(frames[-1, radius_cells:, Ny // 2]).max() print(f"peak |Ez| near the grid edge at the final frame: {outer_ring_energy:.4f} (radiation has arrived)") .. rst-class:: sphx-glr-script-out .. code-block:: none peak |Ez| near the grid edge at the final frame: 0.5455 (radiation has arrived) .. GENERATED FROM PYTHON SOURCE LINES 77-91 The radiation pattern at fixed radius, all the way around the antenna --------------------------------------------------------------------------- The animation only shows the field along the grid; sampling the final ``Ez`` frame on a circle of fixed radius around the source, at every angle, shows the actual radiation *pattern* directly. This 2D TMz-mode soft source is a scalar point emitter, not a true 3D current-element dipole with directional lobes, so on a fine grid it would radiate isotropically; at the modest resolution used here (about 10 grid cells per wavelength), however, the square Yee grid's own numerical dispersion is direction-dependent -- waves travel at a very slightly different phase speed along the grid axes than along the diagonals -- so the pattern instead comes out visibly fourfold-symmetric, a direct picture of that well-known low-order FDTD grid anisotropy rather than of any antenna directionality. .. GENERATED FROM PYTHON SOURCE LINES 91-105 .. code-block:: Python theta = np.linspace(0.0, 2.0 * np.pi, 360, endpoint=False) i_theta = np.clip(np.round(Nx // 2 + radius_cells * np.cos(theta)).astype(int), 0, Nx - 1) j_theta = np.clip(np.round(Ny // 2 + radius_cells * np.sin(theta)).astype(int), 0, Ny - 1) pattern = np.abs(frames[-1, i_theta, j_theta]) fig2 = plt.figure() ax2 = fig2.add_subplot(projection="polar") ax2.plot(theta, pattern) ax2.set_title(f"|Ez| at r = {radius_cells} cells, final frame\n(fourfold pattern: Yee-grid numerical anisotropy, not a directional antenna)") fig2.tight_layout() print(f"radiation pattern at fixed radius: min={pattern.min():.4f}, max={pattern.max():.4f}") print("(the fourfold spread reflects FDTD grid anisotropy at this resolution, not true antenna directionality)") .. image-sg:: /api/gallery/fields/electrodynamics/images/sphx_glr_plot_dipole_radiation_002.png :alt: |Ez| at r = 45 cells, final frame (fourfold pattern: Yee-grid numerical anisotropy, not a directional antenna) :srcset: /api/gallery/fields/electrodynamics/images/sphx_glr_plot_dipole_radiation_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none radiation pattern at fixed radius: min=0.0000, max=0.0504 (the fourfold spread reflects FDTD grid anisotropy at this resolution, not true antenna directionality) .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 4.019 seconds) .. _sphx_glr_download_api_gallery_fields_electrodynamics_plot_dipole_radiation.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_dipole_radiation.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_dipole_radiation.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_dipole_radiation.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_