Magnetic Pressure

The MHD equations can be combined with the Ampère- and Faraday-Maxwell equations,

$$\nabla\times{\bf B} = \mu_0\,{\bf j},$$ (7.7)

$$\nabla\times {\bf E} = - \frac{\partial{\bf B}}{\partial t},$$ (7.8)

respectively, to form a closed set. The displacement current is neglected in Equation (7.7) on the reasonable assumption that MHD motions are slow compared to the velocity of light. Equation (7.8) guarantees that $\nabla\cdot{\bf B}=0$ , provided that this relation is presumed to hold initially. Furthermore, the assumption of quasi-neutrality renders the Poisson-Maxwell equation, $\nabla\cdot{\bf E}=\rho/\epsilon_0$ , redundant.

Equations (7.2) and (7.7) can be combined to give the MHD equation of motion:

$$\rho\,\frac{d{\bf V}}{dt} = -\nabla p + \nabla\cdot{\bf T},$$ (7.9)

where

$$T_{ij} = \frac{B_i\,B_j- \delta_{ij} \,B^{\,2}/2}{\mu_0}.$$ (7.10)

Suppose that the magnetic field is approximately uniform, and directed along the $z$ -axis. In this case, the previous equation of motion reduces to

$$\rho\,\frac{d{\bf V}}{dt} = -\nabla\cdot{\bf P},$$ (7.11)

where

$${\bf P} = \left(\begin{array}{ccc} p + B^{\,2}/2\,\mu_0, & 0, &0\... ... B^{\,2}/2\,\mu_0,& 0\\ [0.5ex] 0,&0,& p - B^{\,2}/2\,\mu_0\end{array} \right).$$ (7.12)

It can be seen that the magnetic field increases the plasma pressure, by an amount $B^{\,2}/(2\,\mu_0)$ , in directions perpendicular to the magnetic field, and decreases the plasma pressure, by the same amount, in the parallel direction. Thus, the magnetic field gives rise to a magnetic pressure, $B^{\,2}/(2\,\mu_0)$ , acting perpendicular to field-lines, and a magnetic tension, $B^{\,2}/(2\,\mu_0)$ , acting along field-lines. Because, as will become apparent in the next section, the plasma is tied to magnetic field-lines, it follows that magnetic field-lines embedded in an MHD plasma act rather like mutually repulsive elastic bands.