Categories: Electromagnetism, Physics, Plasma physics.

# Lorentz force

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

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

\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 $$\vb{u}_0$$:

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

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

\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 $$\vb{E}$$ accelerates the particle with a constant force $$\vb{F} = q \vb{E}$$. Note that the direction depends on the sign of $$q$$.

## Uniform magnetic field

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

\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 $$\vb{u}$$:

\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 $$t$$, and substituting $$\dv*{u_y}{t}$$ from the second, we arrive at the following harmonic oscillator:

\begin{aligned} \dv{u_x}{t} = - \omega_c^2 u_x \end{aligned}

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

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

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

\begin{aligned} u_x(t) = u_\perp \cos\!(\omega_c t) \end{aligned}

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

\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 $$\omega_c$$.

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

\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 $$r_L$$, given by:

\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 $$z$$-axis velocity $$u_z$$, which is conserved:

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

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

\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 $$\vb{x}_{gc}(t) \equiv (x_{gc}, y_{gc}, z_{gc})$$ is the position of the guiding center. For a detailed look at how $$\vb{B}$$ and $$\vb{E}$$ 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.