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+---
+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.