---
title: "Lorentz force"
firstLetter: "L"
publishDate: 2021-09-08
categories:
- Physics
- Electromagnetism
- Plasma physics

date: 2021-09-08T17:00:32+02:00
draft: false
markup: pandoc
---

# Lorentz force

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

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

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.