= 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 are
$$
\lambda_D=\sqrt{\frac{\epsilon_0k_BT_e}{n_ee^2}},\qquad
\omega_{pe}=\sqrt{\frac{n_ee^2}{\epsilon_0m_e}}.
$$
A 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> are
$$
\Omega_c=\frac{|q|B}{m},\qquad r_L=\frac{mv_\perp}{|q|B}.
$$
When 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 decomposition
$$
\mathbf J\times\mathbf B=-\boldsymbol\nabla\left(\frac{B^2}{2\mu_0}\right)
+\frac{(\mathbf B\cdot\boldsymbol\nabla)\mathbf B}{\mu_0}.
$$
These terms are <magnetic pressure> and <magnetic tension>. Magnetic tension resists bending, while magnetic-pressure gradients resist nonuniform compression. The <plasma beta> $\beta=2\mu_0p/B^2$ compares thermal and magnetic stresses.
With uniform <magnetic diffusivity> $\eta_m$, the <resistive induction equation> is
$$
\partial_t\mathbf B=\boldsymbol\nabla\times(\mathbf v\times\mathbf B)+\eta_m\nabla^2\mathbf B.
$$
If the <magnetic Reynolds number> $R_m=UL/\eta_m\gg1$, <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>.
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