---
title: "Lorentz force"
sort_title: "Lorentz force"
date: 2021-09-08
categories:
- Physics
- Electromagnetism
- Plasma physics
layout: "concept"
---

The **Lorentz force** is an empirical force used to define
the [electric field](/know/concept/electric-field/) $$\vb{E}$$
and [magnetic field](/know/concept/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 $$\idv{u_y}{t}$$ from the second,
we arrive at the following harmonic oscillator:

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

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](/know/concept/guiding-center-theory/).



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