Superposition of Discrete Sources

Suppose each of the $N$ students in the class is a separate charge. What is the electrostatic potential in the room due to these charges?

The electrostatic potential due to a single charge is \begin{equation} V = \frac{1}{4\pi\epsilon_0} \> \frac{q}{r} \end{equation} where the $r$ in the denominator is the distance between the source and the observation point.

The superposition principle states that the electrostatic (or gravitational) potential due to a number of static charges (or masses) is simply the sum of the potentials due to the individual sources. This is an extremely important, experimentally-observed law. If the potential due to a number of charges were some complicated non-linear function of the individual potentials or if it depended in some more complicated way on the positions of the sources, it would have been very difficult for us ever to have developed the science of physics.

If there are multiple discrete sources, then there are also many different $r$'s in the denominators of the individual potentials. We can use the results of § {The Distance between Two Objects} to replace each $r$ by the distance between the relevant source and the observation point $|\rr-\rr_i|$. Notice that we have replaced the notation $\rrp$ by $\rr_i$ where the index $i$ is used to distinguish the different sources. The superposition principle for static charges can thus be written \begin{eqnarray*} V(\rr) &= &\frac{1}{4\pi\epsilon_0} \sum_i \frac{q_i}{|\rr-\rr_i|} \end{eqnarray*} and its gravitational counterpart \begin{eqnarray*} \Phi(\rr) &=& -G \sum_i \frac{m_i}{|\rr-\rr_i|} \end{eqnarray*}

In the language of scalar fields, the scalar field representing the potential due to all of the charges is just the (pointwise) sum of the scalar fields representing the potentials due to each of the individual charges.

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