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. $\overrightarrow{a}\times \overrightarrow{b}$ (cross(a, b)) is the cross product between $\overrightarrow{a}$ and $\overrightarrow{b}$.

2. $\Vert \overrightarrow{a}\Vert$ (norm(a)) is the Euclidean norm of $\overrightarrow{a}$.

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_)

Lorentz force via electric and magnetic fields, and velocity.

Law:

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

Latex:

$$\overrightarrow{F}=q(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B})$$

Parameters:

• electric_field_ –

  vector of electric field

  Symbol: E

  Latex: $\overrightarrow{E}$

  Dimension: voltage / length

• magnetic_flux_density_ –

  pseudovector of magnetic flux density

  Symbol: B

  Latex: $\overrightarrow{B}$

  Dimension: magnetic density

• velocity_ –

  vector of particle’s velocity

  Symbol: v

  Latex: $\overrightarrow{v}$

  Dimension: velocity

Returns:

Lorentz force acting on the charged particle

Symbol: F

Latex: $\overrightarrow{F}$

Dimension: force

electric_field_law(lorentz_force_, magnetic_flux_density_, velocity_)

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

Law:

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

Latex:

$$\overrightarrow{E}=\frac{\overrightarrow{F}}{q}-\overrightarrow{v}\times \overrightarrow{B}$$

Parameters:

• lorentz_force_ –

  Lorentz force acting on the charged particle

  Symbol: F

  Latex: $\overrightarrow{F}$

  Dimension: force

• magnetic_flux_density_ –

  pseudovector of magnetic flux density

  Symbol: B

  Latex: $\overrightarrow{B}$

  Dimension: magnetic density

• velocity_ –

  vector of particle’s velocity

  Symbol: v

  Latex: $\overrightarrow{v}$

  Dimension: velocity

Returns:

vector of electric field

Symbol: E

Latex: $\overrightarrow{E}$

Dimension: voltage / length

charge_law(lorentz_force_, electric_field_, magnetic_flux_density_, velocity_)

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

Law:

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

Latex:

$$|q|=\frac{\Vert \overrightarrow{F}\Vert }{\Vert \overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}\Vert }$$

Parameters:

• lorentz_force_ –

  Lorentz force acting on the charged particle

  Symbol: F

  Latex: $\overrightarrow{F}$

  Dimension: force

• electric_field_ –

  vector of electric field

  Symbol: E

  Latex: $\overrightarrow{E}$

  Dimension: voltage / length

• magnetic_flux_density_ –

  pseudovector of magnetic flux density

  Symbol: B

  Latex: $\overrightarrow{B}$

  Dimension: magnetic density

• velocity_ –

  vector of particle’s velocity

  Symbol: v

  Latex: $\overrightarrow{v}$

  Dimension: velocity

Returns:

magnitude of the test charge

Symbol: q

Dimension: charge