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``Field Lines'' and Flux

In Fig. 17.4 the uniform magnetic field is pictured as a forest of little parallel arrows of equal length, equally spaced. Something like this is always necessary if we want to make a visual representation of $\Vec{B}$, but it leaves a lot to be desired. For instance, a uniform magnetic field has the same magnitude and direction at every point in space, not just where the lines are drawn. Moreover, as we have seen, the magnetic force, if any, is never in the direction of the ``lines of $\Vec{B}$'' but rather perpendicular to them, as shown in Fig. 17.4.

Nevertheless, the visual appeal of such a graphical representation in terms of ``field lines'' is so compelling that a whole description of ${\cal E}$&${\cal M}$ has been developed in terms of them. In that description one speaks of ``lines per unit area'' as a measure of the strength of an electric or magnetic field. The analogy is with hydrodynamics, the flow of incompressible fluids, in which we may actually see ``lines'' of fluid flow if we drop packets of dye in the water.

In fluid dynamics there is actually ``stuff'' flowing, a transfer of mass that has momentum and density. In that context one naturally thinks of the FLUX of material through imaginary surfaces perpendicular to the flow^17.8 and indeed $\Vec{B}$ is sometimes referred to as the magnetic flux per unit (perpendicular) area.

By the same token, if ``lines'' of $\Vec{B}$ pass through a surface of area A normal (perpendicular) to $\Vec{B}$, then we can (and do) talk about the MAGNETIC FLUX $\Phi$ through the surface; $\Phi$ has units of magnetic field times area. If we want, we can turn this around and say that a magnetic field has units of flux per unit area.

Even though we rarely take this ``lines of $\Vec{B}$'' business literally, it makes such a good image that we make constant use of it in handwaving arguments. Moreover, the concept of MAGNETIC FLUX is well ensconced in modern ${\cal E}$&${\cal M}$ terminology.