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.