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Unformatted text preview: Chapter 32 Calculating the Electric Field from the Electric Potential 284 32 Calculating the Electric Field from the Electric Potential The plan here is to develop a relation between the electric field and the corresponding electric potential that allows you to calculate the electric field from the electric potential. The electric field is the force-per-charge associated with empty points in space that have a force- per-charge because they are in the vicinity of a source charge or some source charges. The electric potential is the potential energy-per-charge associated with the same empty points in space. Since the electric field is the force-per-charge, and the electric potential is the potential energy-per-charge, the relation between the electric field and its potential is essentially a special case of the relation between any force and its associated potential energy. So, Im going to start by developing the more general relation between a force and its potential energy, and then move on to the special case in which the force is the electric field times the charge of the victim and the potential energy is the electric potential times the charge of the victim. The idea behind potential energy was that it represented an easy way of getting the work done by a force on a particle that moves from point A to point B under the influence of the force. By definition, the work done is the force along the path times the length of the path. If the force along the path varies along the path, then we take the force along the path at a particular point on the path, times the length of an infinitesimal segment of the path at that point, and repeat, for every infinitesimal segment of the path, adding the results as we go along. The final sum is the work. The potential energy idea represents the assignment of a value of potential energy to every point in space so that, rather than do the path integral just discussed, we simply subtract the value of the potential energy at point A from the value of the potential energy at point B. This gives us the change in the potential energy experienced by the particle in moving from point A to point B. Then, the work done is the negative of the change in potential energy. For this to be the case, the assignment of values of potential energy values to points in space must be done just right. For things to work out on a macroscopic level, we must ensure that they are correct at an infinitesimal level. We can do this by setting: Work as Change in Potential Energy = Work as Force-Along-Path times Path Length s d F h h = dU where: dU is an infinitesimal change in potential energy, F h is a force, and s d h is the infinitesimal displacement-along-the-path vector....
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