### Potentials from Continuous Charge Distributions

Suppose that, instead of each of the students in the class being distinguishable, separate charges $q_i$, we instead pack many students together closely so that it makes more sense to idealize the charges in the room as a smoothed out charge density $\rho(\rr)$. What is the electric potential in the room due to this charge density?

We chop the charge density up into $N$ small pieces $\delta\tau_i$, centered at $\rr_i$, and each small enough the the charge density inside each piece is approximately constant, then the superposition principle becomes: \begin{eqnarray} V(\rr) &=& \sum\limits_{i=1}^N {1\over 4\pi\epsilon_0} {q_i\over|\rr-\rr_i|} \nonumber\\ &\approx& \sum\limits_{i=1}^N {1\over 4\pi\epsilon_0} {\rho(\rr_i)\, \delta\tau_i\over|\rr-\rr_i|}\\ &\rightarrow& \int\limits_{\hbox{all charge}} {1\over 4\pi\epsilon_0} {\rho(\rrp)\, \delta\tau'\over|\rr-\rrp|} \nonumber\\ \label{potint} \end{eqnarray} where in the last line the sum becomes an integral in the limit as we chop the charge density finer and finer.