5.13: Electric Potential Field due to a Continuous Distribution of Charge
- Page ID
- 24260
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The electrostatic potential field at \({\bf r}\) associated with \(N\) charged particles is
\[V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^N { \frac{q_n}{\left|{\bf r}-{\bf r}_n\right|} } \label{m0065_eCountable} \]
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
\[q_n = \rho_l({\bf r}_n)~\Delta l \nonumber \]
where \(\rho_l\) is the line charge density (units of C/m) at \({\bf r}_n\). Substituting this expression into Equation \ref{m0065_eCountable}, we obtain
\[{\bf V}({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{\rho_l({\bf r}_n)}{\left|{\bf r}-{\bf r}_n\right|} \Delta l} \nonumber \]
Taking the limit as \(\Delta l\to 0\) yields:
\[V({\bf r}) = \frac{1}{4\pi\epsilon} \int_{\mathcal C} { \frac{\rho_l(l)}{\left|{\bf r}-{\bf r}'\right|} dl} \label{m0065_eLineCharge} \]
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
\[q_n = \rho_s({\bf r}_n)~\Delta s \nonumber \]
where \(\rho_s\) is surface charge density (units of C/m\(^2\)) at \({\bf r}_n\). Substituting this expression into Equation \ref{m0065_eCountable}, we obtain
\[V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{\rho_s({\bf r}_n)}{\left|{\bf r}-{\bf r}_n\right|}~\Delta s} \nonumber \]
Taking the limit as \(\Delta s\to 0\) yields:
\[V({\bf r}) = \frac{1}{4\pi\epsilon} \int_{\mathcal S} { \frac{\rho_s({\bf r}')}{\left|{\bf r}-{\bf r}'\right|}~ds} \label{m0065_eSurfCharge} \]
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
\[q_n = \rho_v({\bf r}_n)~\Delta v \nonumber \]
where \(\rho_v\) is the volume charge density (units of C/m\(^3\)) at \({\bf r}_n\). Substituting this expression into Equation \ref{m0065_eCountable}, we obtain
\[V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{\rho_v({\bf r}_n)}{\left|{\bf r}-{\bf r}_n\right|}~\Delta v} \nonumber \]
Taking the limit as \(\Delta v\to 0\) yields:
\[V({\bf r}) = \frac{1}{4\pi\epsilon} \int_{\mathcal V} { \frac{\rho_v({\bf r}')}{\left|{\bf r}-{\bf r}'\right|}~dv} \nonumber \]
where \({\bf r}'\) represents the varying position over \({\mathcal V}\) with integration.