rockyshephear said:
So what does nable dot B mean in words?
It is easiest to visualize the various vector derivative operations in terms of fluid flow.
Imagine we are considering air in dynamic motion within a room. At an instant in time we imagine a vector field expressing the flux of air, F. You may think of F(x) as the density times the air velocity at a given point.
Now the divergence of F, \nabla \bullet F expresses the expansion of air at a given point. To maintain a steady-state positive divergence at a point we would have to have a source of air. Imagine say a speck of frozen air evaporating at that point. (And of course we can have the reverse, a negative divergence with a spec of frozen air growing) We can however allow a transient divergence at a point which must with conservation of the flowing quantity correspond to a change in concentration. We have the classic conservation equation:
\nabla \bullet F = -\frac{d}{dt}\rho
Divergence equals the decrease in density over time if the vector field expresses the current or flux of the quantity.
Similarly one can consider the vorticity of a fluid flow via the curl. Imagine a hurricane. Draw a circle around the hurricane and you'll note that as you trace around this circle the wind is always flowing in the direction you are tracing. Thus there is a net curl inside this circle (crossing any surface bound by the circle. But be careful here, the curl will not be distributed through out the hurricane but rather is concentrated along the eye-wall. Draw a small circle in the storm but off of the eye-wall and the curl inside will be near zero. As you trace along the circle you'll see for a larger part of the circle farther from the eye you get the wind traveling one way but for the smaller part nearer the eye where the wind is reversed you also have stronger winds. These cancel out. You'll see that the source of the curl is actually the dynamic part of the storm where energy is being converted from heat of the ocean to wind.
Finally the easiest to visualize is the gradient of say a scalar density.
\nabla \rho
expresses a vector in the direction of maximum increase of the density with length equal to the rate of increase in that direction. For a smoothly changing density you'll also see that the rate of increase in an arbitrary direction (unit vector u) will be:
u\bullet \nabla \rho
With these expositions you should then take a look at Stokes and Gauss' laws. I in fact was invoking Stokes formula implicitly when describing the curl in terms of paths in the hurricane.
You may find it instructive, with these fluid analogues of the vector operations to then look at
Maxwell's mechanical model. He imagined space filled with little ball bearings (two sizes) which rotated against each other as well as moving about each other. The smaller bearings expressed flowing charges (with relative density expressing charge density) and the rotation of the other expressed magnetic fields. This is of course a model and not to be taken seriously as a physical description of reality. But it gives a good scaffolding to understand the interplay of electric and magnetic fields.
