Lorentz force via electromagnetic field
=======================================

The **Lorentz force law** states that a charged particle moving in an electromagnetic
field experiences a force that depends on the values of the electric field and the
magnetic field.

**Notation:**

#. :math:`\vec a \times \vec b` (:code:`cross(a, b)`) is the cross product between
   :math:`\vec a` and :math:`\vec b`.
#. :math:`\lVert \vec a \rVert` (:code:`norm(a)`) is the Euclidean norm of :math:`\vec a`.
#. :math:`|x|` (:code:`abs(x)`) is the absolute value of :math:`x`.

**Notes:**

#. This law is valid even in the relativistic case.
#. This law works only in principle because a real particle would generate its own
   electromagnetic field that would interact with the external one which would alter
   the electromagnetic force it experiences.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Lorentz_force>`__.

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

.. py:data:: charge

    Value of the electric :attr:`~symplyphysics.symbols.electrodynamics.charge` of the test particle.

Symbol:
    :code:`q`

Latex:
    :math:`q`

Dimension:
    :code:`charge`


.. py:function:: lorentz_force_law(electric_field_, magnetic_flux_density_, velocity_)

    Lorentz force via electric and magnetic fields, and velocity.

    Law:
        :code:`F = q * (E + cross(v, B))`

    Latex:
        .. math::
            \vec F = q \left( \vec E + \vec v \times \vec B \right)

    :param electric\_field\_: vector of electric field

        Symbol: :code:`E`

        Latex: :math:`\vec E`

        Dimension: *voltage* / *length*

    :param magnetic\_flux\_density\_: pseudovector of magnetic flux density

        Symbol: :code:`B`

        Latex: :math:`\vec B`

        Dimension: *magnetic density*

    :param velocity\_: vector of particle's velocity

        Symbol: :code:`v`

        Latex: :math:`\vec v`

        Dimension: *velocity*

    :return: Lorentz force acting on the charged particle

        Symbol: :code:`F`

        Latex: :math:`\vec F`

        Dimension: *force*


.. py:function:: electric_field_law(lorentz_force_, magnetic_flux_density_, velocity_)

    Electric field via Lorentz force, magnetic field, and velocity.

    Law:
        :code:`E = F / q - cross(v, B)`

    Latex:
        .. math::
            \vec E = \frac{\vec F}{q} - \vec v \times \vec B

    :param lorentz\_force\_: Lorentz force acting on the charged particle

        Symbol: :code:`F`

        Latex: :math:`\vec F`

        Dimension: *force*

    :param magnetic\_flux\_density\_: pseudovector of magnetic flux density

        Symbol: :code:`B`

        Latex: :math:`\vec B`

        Dimension: *magnetic density*

    :param velocity\_: vector of particle's velocity

        Symbol: :code:`v`

        Latex: :math:`\vec v`

        Dimension: *velocity*

    :return: vector of electric field

        Symbol: :code:`E`

        Latex: :math:`\vec E`

        Dimension: *voltage* / *length*


.. py:function:: charge_law(lorentz_force_, electric_field_, magnetic_flux_density_, velocity_)

    Magnitude of the particle's charge via force and electromagnetic field.

    Law:
        :code:`abs(q) = norm(F) / norm(E + cross(v, B))`

    Latex:
        .. math::
            |q| = \frac{\lVert \vec F \rVert}{\left \lVert \vec E + \vec v \times \vec B \right \rVert}

    :param lorentz\_force\_: Lorentz force acting on the charged particle

        Symbol: :code:`F`

        Latex: :math:`\vec F`

        Dimension: *force*

    :param electric\_field\_: vector of electric field

        Symbol: :code:`E`

        Latex: :math:`\vec E`

        Dimension: *voltage* / *length*

    :param magnetic\_flux\_density\_: pseudovector of magnetic flux density

        Symbol: :code:`B`

        Latex: :math:`\vec B`

        Dimension: *magnetic density*

    :param velocity\_: vector of particle's velocity

        Symbol: :code:`v`

        Latex: :math:`\vec v`

        Dimension: *velocity*

    :return: magnitude of the test charge

        Symbol: :code:`q`

        Dimension: *charge*