 Solution

A plasma contains mobile charged species whose collective electromagnetic response matters. On scales large compared with the Debye length, positive and negative charge densities nearly cancel: this is quasineutrality, not the absence of electric currents or electric fields. For electrons of temperature $T_e$ and number density $n_e$, the characteristic plasma screening and oscillation scales areA collective plasma description requires many particles within a screening volume; long-wavelength, sufficiently slow behavior can often be described by magnetohydrodynamics with an appropriate fluid closure.

An individual particle feels the Lorentz force $q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$. In a uniform magnetic field, its transverse motion gyrates while its parallel motion remains free, giving a helical path when both components are present. The nonrelativistic cyclotron frequency and gyroradius areWhen the system scale is much larger than $r_L$ and the relevant time scale longer than ${\Omega }_c^{-1}$, the field strongly organizes transport across it, provided collisions do not interrupt the gyromotion. A perpendicular electric field also gives the common guiding-centre drift ${\mathbf{v}}_E=\mathbf{E}\times \mathbf{B}/B^2$.

For a conducting fluid, the macroscopic Lorentz force density has the useful decompositionThese terms are magnetic pressure and magnetic tension. Magnetic tension resists bending, while magnetic-pressure gradients resist nonuniform compression. The plasma beta $\beta =2{\mu }_{0}p/B^2$ compares thermal and magnetic stresses.

With uniform magnetic diffusivity ${\eta }_m$, the resistive induction equation isIf the magnetic Reynolds number $R_m=UL/{\eta }_m\gg 1$, ideal magnetohydrodynamics applies approximately and magnetic flux freezing links the field geometry to fluid motion. Finite diffusivity permits field slippage and magnetic reconnection. Field tension supports Alfvén waves with $v_A=B/\sqrt{{\mu }_0\rho }$; coupled magnetic and gas compression also supports magnetosonic waves. These effects allow a field to transmit forces and angular momentum between distant parts of an astrophysical plasma.