Motion of magnetic lines of force

Abstract

It is often said that magnetic lines of force in a conducting fluid move with the fluid. In the case of a plasma, this means that the lines of force move with the particle drift velocity 1 v p = ( E × H) H 2 . Such statements are not directly verifiable, since the velocity of a line of force is not a measurable quantity. However, the statement that the lines of force move with a certain velocity v(r, t) does have verifiable consequences, such as: (1) The flux through a closed curve moving with velocity v is constant. (2) A line, moving with velocity v, which is initially a line of force, remains a line of force in the course of its motion. Statement (2), as well as all other verifiable consequences of the original hypothesis, follows from statement (1). A velocity is said to be flux-preserving if it satisfies (1), line-preserving if it satisfies (2). It is permissible to ascribe a velocity v to the lines of force if and only if ∇ × (E + v × H) vanishes identically. It is always possible to choose a v satisfying this relation, although it is not generally possible to do this uniquely. To say that a certain velocity v is permissible means that all the verifiable consequences of ascribing this velocity to the lines of force are valid, i.e., that v is flux-preserving. In the case of the particle drift velocity the condition for flux-preservation reduces to ∇ × [ H(E·H) H 2 ] = 0 . Even if v p is not flux-preserving, there may be some closed curves moving with velocity v p which have constant flux. A semiexhaustive enumeration of such curves is given for a general electromagnetic field. Among these curves are those which lie in a surface everywhere perpendicular to H, if this surface is independent of time. A family of such surfaces will exist if and only if H·∇ × H and H × Ḣ both vanish identically. A velocity may be line-preserving without being flux-preserving, but not vice versa. The necessary and sufficient condition for line-preservation is that H × [∇ × (E + v × H)] should vanish identically. The motion of the lines of force in a plasma is related only to the transverse motion of the charged particles. The latter is separable from the longitudinal motion if and only if v p is line-preserving. A necessary and sufficient condition is also given for the separability of only one component of the transverse motion. The concept of a line of force is not relativistically covariant, because each point of a line of force has the same time coordinate. A curve in space-time which appears as a line of force in one frame of reference will therefore not be a line of force in another frame of reference. However, a moving line of force will trace out a two-dimensional surface in space-time, and it may be that this surface will intersect every space-like hyperplane in a line of force. In that case the surface will appear as the path of a moving line of force in every frame of reference, thus defining a moving line of force as a covariant concept. It is shown that a family of such surfaces exists if and only if E·H vanishes identically, in which case they will be generated by lines of force moving with the particle drift velocity v p .