Understanding the differences and applications of some electrical equations In preparing for the AP Physics 2 test tomorrow, (the first year this test is being offered, so this material is not easily searchable), I've been doing some reviewing of the equation sheet and trying to clear up some difficulties I've always had.
In the electricity and magnetism section of the provided formula sheet, there are several equations near the top that I have always had trouble in understanding. Many of them seem like the same thing, and I have trouble understanding which ones to use when.
Specifically, 
$|F_E| = k {|q_1q_2| \over r^2}$ 
I know this to be Coulombs law, but I always have trouble understanding when to apply it, especially since there is another nearly idnentical equation provided underneath:
$|E| = k {|q| \over r^2}$
Complicating things further, there are more that are similar:
$ V = k {q \over r} $
$ |E| = |{\Delta V \over \Delta r}| $
What are the differences between these equations, and when should each be applied / not applied?
 A: Your first two equations are effectively the same once you consider that $F = q E$. You should use the second one if you're concerned about the electric field in general, but the first one if you want to know what actually happens to a particle (i.e., what force a particle feels). When in doubt, you can usually do alright by working out first the field, and then applying it to the particle whose motion concerns you.
The second set of equations are not really the same thing. One is a definition of voltage, a scalar quantity. Remember, positive charges travel towards regions of low voltage. Voltage just tells you how much energy a charge would have if placed at a spot--that's why it has units of Joules/Coulomb. The second equation tells you what electric field you need to set up a voltage (or, alternately, if you have a straight-line path between voltages, what the electric field would be between them). My guess is the first would be used in a situation like "what is the voltage due to a given charge distribution" while the second is more likely to show up in a problem with capacitors, like "these capacitors are at $\pm V$, find the electric field between them".
