5.9: Independence of Path

In Section 5.8, we found that the potential difference (“voltage”) associated with a path \({\mathcal C}\) in an electric field intensity \({\bf E}\) is \[V_{21} = - \int_{\mathcal C} {\bf E} \cdot d{\bf l} \nonumber \]

where the curve begins at point 1 and ends at point 2. Let these points be identified using the position vectors \({\bf r}_1\) and \({\bf r}_2\), respectively (see Section 4.1). Then:

\[V_{21} = - \int_{\mathbf{r}_{1}, \: \text{along} \: \mathcal{C}}^{\mathbf{r}_{2}} \mathbf{E} \cdot d \mathbf{l} \nonumber \]

The associated work done by a particle bearing charge \(q\) is

\[W_{21} = qV_{21} \nonumber \]

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

\[W_{21} = W_2 - W_1 \nonumber \]

where \(W_2\) and \(W_1\) are the potential energies when the particle is at \({\bf r}_2\) and \({\bf r}_1\), respectively. It is clear from the above equation that \(W_{21}\) does not depend on \({\mathcal C}\); it depends only on the positions of the start and end points and not on any of the intermediate points along \({\mathcal C}\). That is,

\[\boxed{ V_{21} = - \int_{{\bf r}_1}^{{\bf r}_2} {\bf E} \cdot d{\bf l} ~~~\mbox{, independent of}~\mathcal{C} } \label{m0062_eV12a} \]

Since the result of the integration in Equation \ref{m0062_eV12a} is independent of the path of integration, any path that begins at \({\bf r}_1\) and ends at \({\bf r}_2\) yields the same value of \(W_{21}\) and \(V_{21}\). We refer to this concept as independence of path.

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 \ref{m0062_eV12a} using some path other than the path actually traversed.