Introduction to electric potential energy (calculus based physics version).
00:00 Introduction: in this introduction to electric potential energy (calculus based physics version), we start by quickly reviewing the basic definition of work as a path integral and the definition of potential energy for a path-independent (conservative) force. We review how to compute potential energies by using a known conservative force, and we apply this idea to finding the potential energy for a uniform electric field and the potential energy between two point charges. Finally, we compute the potential energy associated with a distribution of many charges.
01:14 Chapter 1: quick review of work and potential energy. We give a quick reminder of work as a path integral and the fact that the integral can be written with and F*dl inside or as Fcos(phi)dl if we want to use the angle between the force vector and the path increment. We also include a reminder of the definition of a conservative force: a force for which the work integral is path independent. Next, we introduce the potential energy as a book-keeping device for computing work: if the work integral is path-independent, then it only depends on initial and final position. So work integrals for conservative forces can be analyzed by comparing the initial and final values of a potential energy function that depends only on initial and final position. Finally we give a reminder that the work done by a conservative force is the negative of the change in potential energy, and we give an example from gravitation to make that minus sign more intuitive.
04:09 Chapter 2: strategy for computing electric potential energies. We review the basic strategy for computing the potential energy associated with a conservative force: compute the work integral and express the result as the negative of a change in a quantity. That quantity is the potential energy function!
05:40 Chapter 3: electric potential energy for a uniform field. We compute the potential energy associated with a uniform electric field in two different ways. First, we assume the work is path independent (in other words, we assume the electric force is conservative). This allows us to choose a simpler path for the integral and we obtain an electric potential energy of qEy. Second, we use the geometry of the dot product to integrate along a curved path and arrive at the same result, showing that the result of the work integral does not depend on the path, but only on the initial and final y values.
11:00 Chapter 4: electric potential energy between two point charges. We compute the potential energy between two point charges in two different ways: first we assume the force is path independent, then choose a clever path from the starting point to the finishing point so that the dot product in the work integral becomes trivial along each part of the path integral (one part of the path is on an arc and the other is radial). We compute the work integral on this path and arrive at the potential energy function 1/(4pi*epsilon_0)*qq_0/r. Second, we show using the geometry of the dot product that the work integral comes out the same, even on an arbitrary path from the starting point to the finishing point. Finally, we compare the graphs of the electric potential energy for two point charges for the cases of like and unlike charges. We also show that we have implicitly chosen a reference point for the potential energy already: U(r) is zero when r goes to infinity. This will matter later when we work with electric potential in the next video!
18:01 Chapter 5: electric potential energy for a collection of many charges. We examine two different cases for the potential energy of a collection of charges. First, we compute the total electric potential energy for a collection of n point charges. The potential energies have to be carefully added up pairwise with no double counting, and we express the total potential energy in summation notation. Second, we compute the potential energy contribution given by introducing a new charge to a collection of existing charges and express that contribution in summation notation. This formula plays a major role in some of our electric potential calculations in the next couple videos.
