Magnetic field due to constant filamentary current
==================================================

Known as the **Biot—Savart law**, it is an equation describing the magnetic field due to a constant
electric current.

**Notation:**

#. :math:`\mu_0` (:code:`mu_0`) is :attr:`~symplyphysics.quantities.vacuum_permeability`.

**Notes:**

#. This version of the law deals with a current in an infinitely thin wire. For a conductor of a
   finite thickness, the following relation must be used:

   .. math::

       I d \vec{\ell} = \vec{J} dV

#. To find the total magnetic flux density, calculate the line integral over the whole contour.

**Conditions:**

#. The system is in a vacuum.

**Links:**

#. `Wikipedia — Biot—Savart law <https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law>`__.

.. py:currentmodule:: symplyphysics.laws.electricity.vector.magnetic_field_due_to_constant_filamentary_current

.. py:data:: magnetic_flux_density_change

    Infinitesimal change of the :attr:`~symplyphysics.symbols.electrodynamics.magnetic_flux_density` at a given point in space.

Symbol:
    :code:`dB`

Latex:
    :math:`d \vec{B}`

Dimension:
    :code:`magnetic_density`


.. py:data:: absolute_permeability

    :attr:`~symplyphysics.symbols.electrodynamics.absolute_permeability` of the medium.

Symbol:
    :code:`mu`

Latex:
    :math:`\mu`

Dimension:
    :code:`inductance/length`


.. py:data:: current

    :attr:`~symplyphysics.symbols.electrodynamics.current` in the contour.

Symbol:
    :code:`I`

Latex:
    :math:`I`

Dimension:
    :code:`current`


.. py:data:: position_vector

    Position vector of the point at which the magnetic flux density is measured. Also see
    :attr:`~symplyphysics.symbols.classical_mechanics.distance_to_origin`.

Symbol:
    :code:`r`

Latex:
    :math:`{\vec r}`

Dimension:
    :code:`length`


.. py:data:: contour_element_position_vector

    Position vector of a point on the integration path. Also see :attr:`~symplyphysics.symbols.classical_mechanics.distance_to_origin`.

Symbol:
    :code:`l`

Latex:
    :math:`\vec{\ell}`

Dimension:
    :code:`length`


.. py:data:: contour_element_displacement

    A vector along the integration path whose magnitude is the length of the differential element in
    the direction of conventional current. Also see :attr:`~symplyphysics.symbols.classical_mechanics.euclidean_distance`.

Symbol:
    :code:`dl`

Latex:
    :math:`d \vec{\ell}`

Dimension:
    :code:`length`


.. py:data:: law

    :code:`dB = mu / (4 * pi) * I * cross(dl, r - l) / norm(r - l)^3`


    Latex:
        .. math::
            d \vec{B} = \frac{\mu}{4 \pi} \frac{I \left[ d \vec{\ell}, {\vec r} - \vec{\ell} \right]}{\left \Vert {\vec r} - \vec{\ell} \right \Vert^{3}}