Independence of Path

In Section 5.8, we found that the potential difference (“voltage”) associated with a path

in an electric field intensity

is

where the curve begins at point 1 and ends at point 2. Let these points be identified using the position vectors

and

, respectively (see Section 4.1).¹ Then:

The associated work done by a particle bearing charge

is

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

and

are the potential energies when the particle is at

and

, respectively. It is clear from the above equation that

does not depend on

; it depends only on the positions of the start and end points and not on any of the intermediate points along

. That is,

Since the result of the integration in Equation 5.9.5 is independent of the path of integration, any path that begins at

and ends at

yields the same value of

and

. We refer to this concept as independence of path.

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 5.9.5 using some path other than the path actually traversed.

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  See Section 4.1 for a refresher on this concept.