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

$V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^N { \frac{q_n}{\left|{\bf r}-{\bf r}_n\right|} } \label{m0065_eCountable}$

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

$q_n = \rho_l({\bf r}_n)~\Delta l$

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

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

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

$q_n = \rho_s({\bf r}_n)~\Delta s$

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

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

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

$q_n = \rho_v({\bf r}_n)~\Delta v$

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

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

Taking the limit as $$\Delta v\to 0$$ yields:

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

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