Skip to main content
Physics LibreTexts

5.9: Independence of Path

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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.

    This page titled 5.9: Independence of Path is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .