5.13 Electric Potential Field due to a Continuous Distribution of Charge

The electrostatic potential field at \({\bf r}\) associated with \(N\) charged particles is

\begin{equation} V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^N { \frac{q_n}{\left|{\bf r}-{\bf r}_n\right|} } \tag{5.104}\end{equation}

where \(q_n\) and \({\bf r_n}\) are the charge and position of the \(n^{\mbox{th}}\) particle. However, it is more common to have a continuous distribution of charge as opposed to a countable number of charged particles. We now consider how to compute \(V({\bf r})\) three types of these commonly-encountered distributions. Before beginning, it’s worth noting that the methods will be essentially the same, from a mathematical viewpoint, as those developed in Section 5.4; therefore, a review of that section may be helpful before attempting this section.

Continuous Distribution of Charge Along a Curve. Consider a continuous distribution of charge along a curve \(\mathcal{C}\). The curve can be divided into short segments of length \(\Delta l\). Then, the charge associated with the \(n^{\mbox{th}}\) segment, located at \({\bf r}_n\), is

\begin{equation} q_n = \rho_l({\bf r}_n)~\Delta l \tag{5.105}\end{equation}

where \(\rho_l\) is the line charge density (units of C/m) at \({\bf r}_n\). Substituting this expression into Equation 5.104, we obtain

\begin{equation} {\bf V}({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{\rho_l({\bf r}_n)}{\left|{\bf r}-{\bf r}_n\right|} \Delta l} \tag{5.106}\end{equation}

Taking the limit as \(\Delta l\to 0\) yields:

\begin{equation} V({\bf r}) = \frac{1}{4\pi\epsilon} \int_{\mathcal C} { \frac{\rho_l(l)}{\left|{\bf r}-{\bf r}'\right|} dl} \tag{5.107}\end{equation}

where \({\bf r}'\) represents the varying position along \({\mathcal C}\) with integration along the length \(l\).

Continuous Distribution of Charge Over a Surface. Consider a continuous distribution of charge over a surface \(\mathcal{S}\). The surface can be divided into small patches having area \(\Delta s\). Then, the charge associated with the \(n^{\mbox{th}}\) patch, located at \({\bf r}_n\), is

\begin{equation} q_n = \rho_s({\bf r}_n)~\Delta s \tag{5.108}\end{equation}

where \(\rho_s\) is surface charge density (units of C/m\(^2\)) at \({\bf r}_n\). Substituting this expression into Equation 5.104, we obtain

\begin{equation} V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{\rho_s({\bf r}_n)}{\left|{\bf r}-{\bf r}_n\right|}~\Delta s} \tag{5.109}\end{equation}

Taking the limit as \(\Delta s\to 0\) yields:

\begin{equation} V({\bf r}) = \frac{1}{4\pi\epsilon} \int_{\mathcal S} { \frac{\rho_s({\bf r}')}{\left|{\bf r}-{\bf r}'\right|}~ds} \tag{5.110}\end{equation}

where \({\bf r}'\) represents the varying position over \({\mathcal S}\) with integration.

Continuous Distribution of Charge in a Volume. Consider a continuous distribution of charge within a volume \(\mathcal{V}\). The volume can be divided into small cells (volume elements) having area \(\Delta v\). Then, the charge associated with the \(n^{\mbox{th}}\) cell, located at \({\bf r}_n\), is

\begin{equation} q_n = \rho_v({\bf r}_n)~\Delta v \tag{5.111}\end{equation}

where \(\rho_v\) is the volume charge density (units of C/m\(^3\)) at \({\bf r}_n\). Substituting this expression into Equation 5.104, we obtain

\begin{equation} V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{\rho_v({\bf r}_n)}{\left|{\bf r}-{\bf r}_n\right|}~\Delta v} \tag{5.112}\end{equation}

Taking the limit as \(\Delta v\to 0\) yields:

\begin{equation} V({\bf r}) = \frac{1}{4\pi\epsilon} \int_{\mathcal V} { \frac{\rho_v({\bf r}')}{\left|{\bf r}-{\bf r}'\right|}~dv} \tag{5.113}\end{equation}

where \({\bf r}'\) represents the varying position over \({\mathcal V}\) with integration.