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:

  1. \(\vec a \times \vec b\) (cross(a, b)) is the cross product between \(\vec a\) and \(\vec b\).

  2. \(\lVert \vec a \rVert\) (norm(a)) is the Euclidean norm of \(\vec a\).

  3. \(|x|\) (abs(x)) is the absolute value of \(x\).

Notes:

  1. This law is valid even in the relativistic case.

  2. 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:

  1. Wikipedia.

charge

Value of the electric charge of the test particle.

Symbol:

q

Latex:

\(q\)

Dimension:

charge

lorentz_force_law(electric_field_, magnetic_flux_density_, velocity_)[source]

Lorentz force via electric and magnetic fields, and velocity.

Law:

F = q * (E + cross(v, B))

Latex:
\[\vec F = q \left( \vec E + \vec v \times \vec B \right)\]
Parameters:

  • electric_field_ –

    vector of electric field

    Symbol: E

    Latex: \(\vec E\)

    Dimension: voltage / length

  • magnetic_flux_density_ –

    pseudovector of magnetic flux density

    Symbol: B

    Latex: \(\vec B\)

    Dimension: magnetic density

  • velocity_ –

    vector of particle’s velocity

    Symbol: v

    Latex: \(\vec v\)

    Dimension: velocity

Returns:

Lorentz force acting on the charged particle

Symbol: F

Latex: \(\vec F\)

Dimension: force

electric_field_law(lorentz_force_, magnetic_flux_density_, velocity_)[source]

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

Law:

E = F / q - cross(v, B)

Latex:
\[\vec E = \frac{\vec F}{q} - \vec v \times \vec B\]
Parameters:

  • lorentz_force_ –

    Lorentz force acting on the charged particle

    Symbol: F

    Latex: \(\vec F\)

    Dimension: force

  • magnetic_flux_density_ –

    pseudovector of magnetic flux density

    Symbol: B

    Latex: \(\vec B\)

    Dimension: magnetic density

  • velocity_ –

    vector of particle’s velocity

    Symbol: v

    Latex: \(\vec v\)

    Dimension: velocity

Returns:

vector of electric field

Symbol: E

Latex: \(\vec E\)

Dimension: voltage / length

charge_law(lorentz_force_, electric_field_, magnetic_flux_density_, velocity_)[source]

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

Law:

abs(q) = norm(F) / norm(E + cross(v, B))

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

  • lorentz_force_ –

    Lorentz force acting on the charged particle

    Symbol: F

    Latex: \(\vec F\)

    Dimension: force

  • electric_field_ –

    vector of electric field

    Symbol: E

    Latex: \(\vec E\)

    Dimension: voltage / length

  • magnetic_flux_density_ –

    pseudovector of magnetic flux density

    Symbol: B

    Latex: \(\vec B\)

    Dimension: magnetic density

  • velocity_ –

    vector of particle’s velocity

    Symbol: v

    Latex: \(\vec v\)

    Dimension: velocity

Returns:

magnitude of the test charge

Symbol: q

Dimension: charge