The gyration (helical rotation) of a particle with charge $q$ within electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields occurs due to the combined effect of Coulomb force and Lorentz force.¹⁾ The equation of motion for this particle can be expressed as:

$$ m\frac{d\mathbf{v}}{dt} = q(\mathbf{E}+\mathbf{v}\times\mathbf{B}) $$

where $m$ is the mass of the particle and $\mathbf{v}$ is its velocity. In the absence of an electric field, only the Lorentz force term remains in this equation. By taking the dot product of both sides with the velocity, we obtain:

$$ m\frac{d\mathbf{v}}{dt} \cdot \mathbf{v} = q (\mathbf{v}\times\mathbf{B}) \cdot \mathbf{v} \ \Rightarrow \frac{d}{dt}\left(\frac{1}{2}mv^2\right) = 0 $$

Since $\mathbf{v}\cdot (\mathbf{v}\times\mathbf{B})=0$ and both sides are divided by 2, it implies that the kinetic energy ($mv^2/2$) and the magnitude of the velocity (speed) remain constant. A static magnetic field can never change the particle’s kinetic energy.

If the magnetic field is aligned along the $z$-axis, $\mathbf{B}=B\hat{k}$, the components of the equation of motion become:

\begin{align*} & m\dot{v}_x = qBv_y \\ & m\dot{v}_y = - qBv_x \\ & m\dot{v}_z = 0 \end{align*}

where $\dot{v}_x=dv_x/dt$ represents the first derivative. The same holds for the other components. The last equation states that the $z$-component of the velocity, parallel to the magnetic field, remains constant. Differentiating the first equation yields the second derivative:

\begin{align*} & m\ddot{v}_x = qB\dot{v}_y = qB(-qBv_x/m) \\ & \Rightarrow \ddot{v}_x = -\left(\frac{qB}{m}\right)^2 v_x = -\omega_g^2 v_x \end{align*}

where $\omega_g=(qB/m)$ is the gyrofrequency or cyclotron frequency, whose sign is opposite for positive and negative charges. Similarly, it can be shown that $\ddot{v}_y = -\omega^2_g v_y$ for the other component. These second-order differential equations are those of a harmonic oscillator, with solutions:

\begin{align} & x - x_0 = r_g \sin\omega_g t \label{ho1} \\ & y - y_0 = r_g \cos\omega_g t \label{ho2} \end{align}

The sine terms in the displacement components will be opposite for electrons and ions. The gyro-radius is given by:

$$ r_g = \frac{v_\perp}{|\omega_g|} = \frac{mv_\perp}{|q|B} $$

where $v_\perp = (v_x^2+v_y^2)^{1/2}$ is the constant speed in the plane perpendicular to the magnetic field.

According to equations $\ref{ho1}$ and $\ref{ho2}$, the particle follows a circular orbit. The center of this orbit, $(x_0,y_0)$, is known as the guiding center. A circular current flows along the orbit, and the internal magnetic field generated by this current opposes the external magnetic field that created it. This phenomenon is known as the diamagnetic effect.

If the particle has no velocity component parallel to the magnetic field, it will continue to move in a circular path. However, if a parallel component exists ($v_z$ or $v_\parallel$), the particle will follow a helical trajectory, as illustrated below.

The distance between successive orbits along the magnetic field direction is called the pitch, usually measured by the pitch angle ($\alpha$):

$$ \alpha = \tan^{-1} \frac{v_\perp}{v_\parallel} $$

Thus, the pitch angle depends on the ratio of the two velocity components.