Exploring V and E in 1-D The mathematics of the potential and the field are much easier to visualize and to calculate when we limit ourselves to one dimension. The next three problems will be based on the following system. A plastic rod of length L has a total charge, Q, distributed evenly along HS length. This means that the a ................. charge per unit length, A = Q/L, is constant. We will be examining the field and potential \ at a pomt a from the end of the rod. Finding V Rod of length L and and E here charge Q Problem 1: Finding V at Point a To find the potential at point a, we must first consider the rod to be made of a bunch of small pieces of length dx. One such piece of the rod located a distance, x, from the end of the rod is shown in the following figure: Consider a piece of the rod that is dx long, and x from the end This piece of the rod is a distance r from a, and has a charge dq: r=a+x = 9 dx L _ 2 dq L dx Use this information to find the voltage generated by this small point charge: Now integrate your value of dV to find the total V at point a: * = L v = [ av *= 0 V= Now generalize your expression. Write out V(x) for any point a distance x to the left of the rod (replace a with x). V(x) =Problem 2: Finding E at Point a Similarly, find dB (the field caused by the piece of the rod of thickness dx located at x) and E (the total field at point a caused by the rod). E = What direction does E point? Now generalize your expression. Write out E(x) for any point a distance x to the left of the rod (replace a with X). E(x) = Post a question that comes up for you about the content on Worksheet 1. 0 Your question may be specic to a given problem or a more general concept or math methods question. 0 If specic to a given problem, please identify which problem you are asking your question about when you post your question. 0 Your question must be unique. While you are encouraged to empathize with others' questions if you have the same one, you must post a unique question to receive credit