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Formula Lorentz Force (Vector Equation) Magnetic field    Electric field

Formula: Lorentz Force (Vector Equation)
Electron movement at an angle to the magnetic field

Lorentz force

Lorentz force is the sum of electric force \(q \, \boldsymbol{E}\) and magnetic force \( q \, \boldsymbol{v} ~\times~ \class{violet}{\boldsymbol{B}} \) on a charged particle with charge \( q \). The Lorentz force is - due to the cross product - orthogonal to velocity \(\boldsymbol{v}\) of the charged particle and to the external magnetic field \(\class{violet}{\boldsymbol{B}}\).

Written out, the cross product between the velocity and the magnetic field reads as follows:\[ \boldsymbol{v} ~\times~ \class{violet}{\boldsymbol{B}} ~=~ \begin{bmatrix} v_y \,B_z - v_z \, B_y \\ v_z \, B_x - v_x \, B_z \\ v_x \, B_y - v_y \, B_x \end{bmatrix} \]


Velocity with which the charged particle moves through the external magnetic field. The greater the speed of the particle, the greater the Lorentz force.

Velocity is a vector with three components:$$ \boldsymbol{v} ~=~ \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} $$

Magnetic field

Magnetic flux density determines the strength of the external magnetic field in which the charged particle moves. The greater the magnetic flux density, the greater the Lorentz force on the particle.

The magnetic flux density is a vector with three components:$$ \class{violet}{\boldsymbol{B}} ~=~ \begin{bmatrix} B_x \\ B_y \\ B_z \end{bmatrix} $$

Electric charge

Electric charge of the particle on which the Lorentz force is exerted. The greater the charge of the particle, the greater the Lorentz force.

Electric field

Electric field indicates what the force on a charged particle would be if it were placed at some location in the electric field. If the E-field is multiplied by the charge \(q\), then the product is the electric force \( q \, \boldsymbol{E} \) on the particle. For example, the E-field may be the homogeneous E-field between two capacitor plates.

The E-field is a vector with three components:\[ \boldsymbol{E} ~=~ \begin{bmatrix} E_x \\ E_y \\ E_z \end{bmatrix} \]