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Potential of a Point Charge and Superposition

We would really love to at least in principle be able to evaluate the potential of an arbitrary charge density distribution without having to first evaluate the field. Suppose we start by evaluating the potential of a point charge at a point $\vec{r}$ in charge-centered coordinates:


$\displaystyle V(\vec{r})$ $\textstyle =$ $\displaystyle - \int_\infty^{\vec{r}} \vec{E} \cdot d\vec{l}$  
  $\textstyle =$ $\displaystyle - \int_\infty^r E_r dr$  
  $\textstyle =$ $\displaystyle - \int_\infty^r k q r^{-2} dr$  
  $\textstyle =$ $\displaystyle \frac{kq}{r}$ (8)

This is just groovy. The potential of a point charge is just $kq$ divided by the distance to the point of observation. If we write this in more general coordinates:


\begin{displaymath}
V(\vec{r}) = \frac{kq}{\left\vert\vec{r} - \vec{r}_0\right\vert}
\end{displaymath} (9)

(where now the charge is at the position $\vec{r}_0$) then we can use the linearity of integration in the definition of $V$ above to write


\begin{displaymath}
V(\vec{r}) = \sum_i \frac{kq_i}{\left\vert\vec{r} - \vec{r}_i\right\vert}.
\end{displaymath} (10)

We are now cooking, in a manner of speaking. With this recipe, we can add up the potential of any discrete number of point charges quite easily. Note that because it is a scalar, we just add up the numbers - we don't have to worry about components per se.

If we smear the point charges out into a charge density distribution $\rho(\vec{r}_0)$, the sum becomes the usual integral:


$\displaystyle V(\vec{r})$ $\textstyle =$ $\displaystyle \int \frac{k dq(\vec{r}_0)}{ \left\vert \vec{r} -
\vec{r}_0 \right\vert }$  
  $\textstyle =$ $\displaystyle \int \frac{k \rho(\vec{r}_0)d^3r_0}{ \left\vert \vec{r} -
\vec{r}_0 \right\vert }$ (11)

where the integral is over all charge in the distribution. This latter result is very important. It is, in fact, the solution to a second order partial differential equation derived from Gauss's Law and the connection between potential and field.

For culture:


next up previous
Next: The Poisson Equation Up: Electrostatic Potential Previous: Potential versus Potential Difference:
Robert G. Brown 2002-01-30