Categories: Electromagnetism, Physics, Plasma physics.

Lorentz force

The Lorentz force is an empirical force used to define the electric field E\vb{E} and magnetic field B\vb{B}. For a particle with charge qq moving with velocity u\vb{u}, the Lorentz force F\vb{F} is given by:

F=q(E+u×B)\begin{aligned} \boxed{ \vb{F} = q (\vb{E} + \vb{u} \cross \vb{B}) } \end{aligned}

Uniform electric field

Consider the simple case of an electric field E\vb{E} that is uniform in all of space. In the absence of a magnetic field B=0\vb{B} = 0 and any other forces, Newton’s second law states:

F=mdudt=qE\begin{aligned} \vb{F} = m \dv{\vb{u}}{t} = q \vb{E} \end{aligned}

This is straightforward to integrate in time, for a given initial velocity vector u0\vb{u}_0:

u(t)=qmEt+u0\begin{aligned} \vb{u}(t) = \frac{q}{m} \vb{E} t + \vb{u}_0 \end{aligned}

And then the particle’s position x(t)\vb{x}(t) is found be integrating once more, with x(0)=x0\vb{x}(0) = \vb{x}_0:

x(t)=q2mEt2+u0t+x0\begin{aligned} \boxed{ \vb{x}(t) = \frac{q}{2 m} \vb{E} t^2 + \vb{u}_0 t + \vb{x}_0 } \end{aligned}

In summary, unsurprisingly, a uniform electric field E\vb{E} accelerates the particle with a constant force F=qE\vb{F} = q \vb{E}. Note that the direction depends on the sign of qq.

Uniform magnetic field

Consider the simple case of a uniform magnetic field B=(0,0,B)\vb{B} = (0, 0, B) in the zz-direction, without an electric field E=0\vb{E} = 0. If there are no other forces, Newton’s second law states:

F=mdudt=qu×B\begin{aligned} \vb{F} = m \dv{\vb{u}}{t} = q \vb{u} \cross \vb{B} \end{aligned}

Evaluating the cross product yields three coupled equations for the components of u\vb{u}:

duxdt=qBmuyduydt=qBmuxduzdt=0\begin{aligned} \dv{u_x}{t} = \frac{q B}{m} u_y \qquad \quad \dv{u_y}{t} = - \frac{q B}{m} u_x \qquad \quad \dv{u_z}{t} = 0 \end{aligned}

Differentiating the first equation with respect to tt, and substituting duy/dt\idv{u_y}{t} from the second, we arrive at the following harmonic oscillator:

d2uxdt2=ωc2ux\begin{aligned} \dvn{2}{u_x}{t} = - \omega_c^2 u_x \end{aligned}

Where we have defined the cyclotron frequency ωc\omega_c as follows, which may be negative:

ωcqBm\begin{aligned} \boxed{ \omega_c \equiv \frac{q B}{m} } \end{aligned}

Suppose we choose our initial conditions so that the solution for ux(t)u_x(t) is given by:

ux(t)=ucos(ωct)\begin{aligned} u_x(t) = u_\perp \cos(\omega_c t) \end{aligned}

Where uux2+uy2u_\perp \equiv \sqrt{u_x^2 + u_y^2} is the constant total transverse velocity. Then uy(t)u_y(t) is found to be:

uy(t)=mqBduxdt=mωcqBusin(ωct)=usin(ωct)\begin{aligned} u_y(t) = \frac{m}{q B} \dv{u_x}{t} = - \frac{m \omega_c}{q B} u_\perp \sin(\omega_c t) = - u_\perp \sin(\omega_c t) \end{aligned}

This means that the particle moves in a circle, in a direction determined by the sign of ωc\omega_c.

Integrating the velocity yields the position, where we refer to the integration constants xgcx_{gc} and ygcy_{gc} as the guiding center, around which the particle orbits or gyrates:

x(t)=uωcsin(ωct)+xgcy(t)=uωccos(ωct)+ygc\begin{aligned} x(t) = \frac{u_\perp}{\omega_c} \sin(\omega_c t) + x_{gc} \qquad \quad y(t) = \frac{u_\perp}{\omega_c} \cos(\omega_c t) + y_{gc} \end{aligned}

The radius of this orbit is known as the Larmor radius or gyroradius rLr_L, given by:

rLuωc=muqB\begin{aligned} \boxed{ r_L \equiv \frac{u_\perp}{|\omega_c|} = \frac{m u_\perp}{|q| B} } \end{aligned}

Finally, it is easy to integrate the equation for the zz-axis velocity uzu_z, which is conserved:

z(t)=zgc=uzt+z0\begin{aligned} z(t) = z_{gc} = u_z t + z_0 \end{aligned}

In conclusion, the particle’s motion parallel to B\vb{B} is not affected by the magnetic field, while its motion perpendicular to B\vb{B} is circular around an imaginary guiding center. The end result is that particles follow a helical path when moving through a uniform magnetic field:

x(t)=uωc(sin(ωct)cos(ωct)0)+xgc(t)\begin{aligned} \boxed{ \vb{x}(t) = \frac{u_\perp}{\omega_c} \begin{pmatrix} \sin(\omega_c t) \\ \cos(\omega_c t) \\ 0 \end{pmatrix} + \vb{x}_{gc}(t) } \end{aligned}

Where xgc(t)(xgc,ygc,zgc)\vb{x}_{gc}(t) \equiv (x_{gc}, y_{gc}, z_{gc}) is the position of the guiding center. For a detailed look at how B\vb{B} and E\vb{E} can affect the guiding center’s motion, see guiding center theory.


  1. F.F. Chen, Introduction to plasma physics and controlled fusion, 3rd edition, Springer.