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 $V_{21} = - \int_{\mathcal C} {\bf E} \cdot d{\bf l}$

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:

$V_{21} = - \int_{\bf r}^1,~\mbox{along}~\mathcal{C}^{\bf r_2} {\bf E} \cdot d{\bf l}$

The associated work done by a particle bearing charge $$q$$ is

$W_{21} = qV_{21}$

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

$W_{21} = W_2 - W_1$

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.