I am being confused by what should be a simple physical scenario. Let's say I have a small, cylindrical region of radius and height are both $a$ filled with a medium of both positive and negative charges - let's say with a conducitivity of $\sigma$. Assume in the radial plane there is a sinusoidal force (for example from any kind of pressure wave) which acts on all the particles (both positive and negative) in the region to create a velocity flow, $v_{pressure} = v_0 sin(\omega t) \hat{r}$ in the radial direction. Also assume there is a constant magnetic field pointing in the orthogonal $\hat{z}$ direction, $\vec{B_{bias}} = B_0 \hat{z}$. The combination of velocity flow and constant magnetic field will create a Lorentz force on the charges, creating a cylindrically symmetric current distribution: \begin{equation} F = q(\vec{v} \times \vec{B_{bias}}) \end{equation} \begin{equation} J = \sigma (\vec{v} \times \vec{B_{bias}}) = v_0 B_0 sin(\omega t) \hat{\phi} \end{equation}
How can I calculate the RF magnetic field that this current will create? The charges in the current will have an acceleration due to 1) the time varying velocity field, $v_{pressure} = v_0 sin(\omega t) \hat{r}$ and 2) the acceleration of the charges moving in curves, $v_{phi}^2 / R$ where $v_{phi}$ is the velocity created by the Lorentz force in the $\hat{\phi}$ direction and $R$ is the radius of current distribution.
To me it seems that I can either use Maxwell's equations \begin{equation} J = \sigma E_{rf} + \sigma \vec{v} \times \vec{B_{bias}} \end{equation} \begin{equation} \nabla \times E_{rf} = - \frac{\partial B_{rf}}{\partial t} \end{equation} \begin{equation} \nabla \times B_{rf} = \mu_0 (J + \epsilon \frac{\partial E_{rf}}{\partial t}) \end{equation} However, I am uncertain about the $E_{rf}$ term. There is no external electric field. The lorentz force drives the cylindrically symmetric current distribution so it alone should be responsible for the current flow, in which case I should not need to add this extra term, is this correct? Is it the case that the Lorentz forces alone creates this cylindrically symmetric current distribution which is changing with time and via Faraday's law this automatically creates a time-varying magnetic field in the \hat{z} direction and I don't need to add this term?
Alternatively, I can calculate the magnetic field via \begin{equation} B_{rf} = \frac{\mu_0}{4 \pi} \frac{q \vec{a} \times \vec{r}}{r^3} \end{equation} where $\vec{a}$ is the acceleration each charge feels and $r$ the direction from the charge to the point of interest.
Can anyone help me set this up please? Thank you in advance.
