Spherical Unit Vectors . The difficult thing to keep straight about these unit vectors is that their direction is a function of where you are in space. That is, they are always about. To explore this idea, click on the icon. In the animated image the r unit vector is in red, the theta unit vector is in green, and the phi unit vector is in blue. At first the value of theta is changing, but you can also choose to let phi change instead. Make sure you have the behavior of these unit vectors clearly in mind.
Cylindrical Unit Vectors . The cylindrical unit vectors are a little simpler. The z unit vector always points in the same direction, so only the r and theta unit vectors change with your position in space. But it is still important to understand how they move, so try pushing the button.
Divergence, This link explores the idea of the divergence of a vector field. Roughly speaking, if the vector field looks like stuff is flowing outward from some source then it has positive divergence, or if it is flowing inward toward some sink then it has negative divergence. Unfortunately, this is only part of the story. A more sophisticated way of seeing what divergence means is to imagine that you are inside a bubble embedded in the vector field and that your are flowing along with it. You then ask the question, "Is the bubble expanding or contracting?" If it is expanding the flow has positive divergence; if it is contracting the flow has negative divergence; and if the bubble volume remains constant the divergence is zero. Even if the bubble changes shape, if its volume is constant the divergence is zero. Click to see examples of a zero-divergence field and a positive divergence field.
Curl, You can get a rough idea of whether a vector field has curl, or not, by looking to see if it "swirls". And you can even get the direction of the curl by curving the fingers of your right hand along the flow and checking the direction of your thumb. But as with divergence, this is not the whole story. A better way to understand curl is to imagine that a piece of confetti has been placed in the flow. You now ask the question, "Does the confetti rotate?" If it does, then the flow has curl, with the direction of the curl gotten by using the right-hand rule on the rotating bit of paper. And if the confetti maintains its orientation without rotating, then the curl is zero. Click below to see examples of both kinds of flows.
Integrating to get E Located behind this virtual wall is an animation of how to use integration to find the electric field along the z-axis of a ring of charge.
Langevin Model of Polarization , what's that? Well, it's the physical process that goes with the horrible mathematics of Problem 4.38 in Griffiths. Click here to see it.
.