5.9: Independence of Path
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In Section 5.8, we found that the potential difference (“voltage”) associated with a path
where the curve begins at point 1 and ends at point 2. Let these points be identified using the position vectors
The associated work done by a particle bearing charge
This work represents the change in potential energy of the system consisting of the electric field and the charged particle. So, it must also be true that
where
Since the result of the integration in Equation
The integral of the electric field over a path between two points depends only on the locations of the start and end points and is independent of the path taken between those points.
A practical application of this concept is that some paths may be easier to use than others, so there may be an advantage in computing the integral in Equation