Potentials from Continuous Charge Distributions

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Section 9.5 Potentials from Continuous Charge Distributions

Suppose that, instead of each of the students in the class being distinguishable, separate charges \(q_i\text{,}\) 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 \(\rho(\rr)\text{.}\) What is the electric potential in the room due to this charge density?

We chop the charge density up into \(N\) small pieces \(\delta\tau_i\text{,}\) centered at \(\rr_i\text{,}\) and each small enough that the charge density inside each piece is approximately constant, then the superposition principle becomes:

\begin{align} V(\rr) \amp= \sum\limits_{i=1}^N {1\over 4\pi\epsilon_0} {q_i\over|\rr-\rr_i|}\notag\\ \amp\approx \sum\limits_{i=1}^N {1\over 4\pi\epsilon_0} {\rho(\rr_i)\, \delta\tau_i\over|\rr-\rr_i|}\tag{9.5.1}\\ \amp\rightarrow \int\limits_{\textrm{all charge}} {1\over 4\pi\epsilon_0} {\rho(\rrp)\, \delta\tau'\over|\rr-\rrp|}\tag{9.5.2} \end{align}

where in the last line the sum becomes an integral in the limit as we chop the charge density finer and finer.