A charged particle in electric and magnetic fields

When a charged particle with a charge $q$ and a mass $m$ moves in a electric field $\vec{E}$ and a magnetic field $\vec{B}$, the force on the particle is,

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

Written out in terms of its three components, the Lorentz force is,

$F_x = q(E_x+v_yB_z-v_zB_y)$,
$F_y = q(E_y+v_zB_x-v_xB_z)$,
$F_z = q(E_z+v_xB_y-v_yB_x).$

In general, the three components of the electric and magnetic fields can be functions of space and time. The force defines the trajectory that the charged particle will follow via Newton's law, $\vec{F}=m\vec{a}$. Newton's law can be written as three coupled second-order differential equations.

$F_x = m\frac{d^2x}{dt^2}= q(E_x(x,y,z,t)+v_yB_z(x,y,z,t)-v_zB_y(x,y,z,t))$,
$F_y = m\frac{d^2y}{dt^2} = q(E_y(x,y,z,t)+v_zB_x(x,y,z,t)-v_xB_z(x,y,z,t))$,
$F_z = m\frac{d^2z}{dt^2} = q(E_z(x,y,z,t)+v_xB_y(x,y,z,t)-v_yB_x(x,y,z,t)).$

The components of the electric and magnetic fields can be specified in the text boxes below these can then be loaded into the general 3-D motion differential equation solver to calculate the trajectory of the charged particle.

When a constant uniform electric field is perpendicular to a constant uniform magnetic field, the average velocity of the charged particle will be in the direction perpendicular to both the electric field and the magnetic field. Note that electrons move very fast so a short time step $\Delta t$ must be selected.