.. _sphx_glr_api_gallery_fields_statics:

Electrostatics and magnetostatics
------------------------------------

Static fields from charges and steady currents: Coulomb's inverse-square
law, the Laplace-Legendre multipole expansion, Poisson's equation solved
on a grid by relaxation and by fast transforms, Green's method of images
for conductors, and the Biot-Savart field of a current loop and a finite
solenoid.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Weeks after Ørsted saw a compass needle swing beside a current-carrying wire in 1820, Jean-Baptiste Biot and Félix Savart measured how the magnetic force fell off with distance from a long wire. Laplace turned their result into a law for each small piece of wire:">

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  .. image:: /api/gallery/fields/statics/images/thumb/sphx_glr_plot_biot_savart_loop_and_solenoid_thumb.png
    :alt:

  :doc:`/api/gallery/fields/statics/plot_biot_savart_loop_and_solenoid`

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      <div class="sphx-glr-thumbnail-title">The Biot-Savart law: a current loop and a finite solenoid</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In 1785 Charles-Augustin de Coulomb hung a charged pith ball from a torsion balance and measured how the twist needed to hold a second charge at bay grew as he pushed them together. He found the electric force falls off as the inverse square of the distance,">

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  .. image:: /api/gallery/fields/statics/images/thumb/sphx_glr_plot_coulomb_inverse_square_law_thumb.png
    :alt:

  :doc:`/api/gallery/fields/statics/plot_coulomb_inverse_square_law`

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      <div class="sphx-glr-thumbnail-title">Coulomb's law: the inverse-square force and superposition</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="George Green&#x27;s 1828 Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism showed that the potential inside any region follows from a single function: the potential of a unit point charge that vanishes on the region&#x27;s boundary. Once you know this Green&#x27;s function, you can find the potential for any charge inside the region and any voltages on its boundary.">

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  .. image:: /api/gallery/fields/statics/images/thumb/sphx_glr_plot_green_method_of_images_thumb.png
    :alt:

  :doc:`/api/gallery/fields/statics/plot_green_method_of_images`

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      <div class="sphx-glr-thumbnail-title">Green's function and the method of images</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Legendre (1782) expanded 1/|\mathbf r - \mathbf r&#x27;| in powers of r&#x27;/r while computing the gravitational pull of a spheroid, and so introduced the polynomials that bear his name. Laplace (1785) extended the idea to the full angular dependence, the spherical harmonics. The result, applied to charges, is the multipole expansion:">

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  .. image:: /api/gallery/fields/statics/images/thumb/sphx_glr_plot_laplace_legendre_multipole_expansion_thumb.png
    :alt:

  :doc:`/api/gallery/fields/statics/plot_laplace_legendre_multipole_expansion`

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      <div class="sphx-glr-thumbnail-title">Laplace and Legendre: the multipole expansion of a far field</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Laplace had shown that the gravitational (and so electrostatic) potential satisfies \nabla^2\phi = 0 in empty space. In 1813 Siméon Denis Poisson showed that inside matter the right-hand side is the source density:">

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  .. image:: /api/gallery/fields/statics/images/thumb/sphx_glr_plot_poisson_equation_relaxation_thumb.png
    :alt:

  :doc:`/api/gallery/fields/statics/plot_poisson_equation_relaxation`

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      <div class="sphx-glr-thumbnail-title">Poisson's equation: relaxing the potential of charges and conductors</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/fields/statics/plot_biot_savart_loop_and_solenoid
   /api/gallery/fields/statics/plot_coulomb_inverse_square_law
   /api/gallery/fields/statics/plot_green_method_of_images
   /api/gallery/fields/statics/plot_laplace_legendre_multipole_expansion
   /api/gallery/fields/statics/plot_poisson_equation_relaxation