Macroscopic cold-fluid equilibrium properties of relativistic non-neutral electron flow in a cylindrical diode

The cold-fluid Maxwell equations are used to investigate detailed equilibrium properties (∂/∂t=0) of relativistic non-neutral electron flow in cylindrical diode geometry with applied magnetic field. There is considerable latitude in specifying the equilibrium profiles. In particular, the profiles for any one of the density $n_b^0$(r), axial magnetic field $B_z^0$(r), radial electric field $E_r^0$(r), or azimuthal flow velocity $V_{\mathrm{thetab}}^0$(r)=${\beta }_b$(r)c can be specified, and the remaining three profiles calculated self-consistently from Poisson’s equation, the ∇×${\mathrm{B}}^0$ Maxwell equation, and the radial force balance equation. As a first example, assuming that the functional form of ${\beta }_b$(r) is specified, formal expressions are derived for $B_z^0$(r), $n_b^0$(r), and $E_r^0$(r). As a second example, it is assumed that the radial electric field is prescribed by $E_r^0$(r)=const×sinh[κ(r-a)] over the radial extent of the electron layer extending from r=a (the cathode) to r=$r_b$ (the outer edge of the electron layer). In this case, a closed-form expression for the density profile $n_b^0$(r) is obtained, but the profiles for the magnetic field $B_z^0$(r) and flow velocity ${\beta }_b$(r)c must generally be determined numerically when the layer aspect ratio A=a/($r_b$-a) is finite. The influence of cylindrical effects (e.g., centrifugal effects) and finite aspect ratio A on detailed equilibrium properties is calculated, treating the planar diode limit (A=∞) as a reference case where the equilibrium profiles are known in closed analytic form. It is found that cylindrical effects and finite aspect ratio can have a large influence on profile shape, particularly when the electron flow is relativistic.