Superposition and the Computer

The principle of superposition is an overarching and powerful tool in much of physics. It is useful well beyond the electric field as you will see with the magnetic field (and as you might see in future physics courses in quantum mechanics). The fact that the electric field obeys the principle of superposition defines a powerful algorithm for computing the electric field any given location from any distribution of charge. In these notes, you will read about how that algorithm works.

As a reminder, the principle of superposition states that the electric field at any given location in space is determined by vector sum of the electric field due to each charge that contributes.

$$\vec{E}_{net} = \sum \vec{E}_i = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \dots$$

You have seen how this principle can be used to find the electric field due to point charges and how it has been used for “continuous charge distributions” like the line charge example. In the line charge example, you sliced up the line into little bits, which each contributed a small amount of electric field $d\vec{E}$ at a given location. The total electric field at that same given location was the integral (continuous sum) of the contributions,

$$\vec{E}_{net} = \int d\vec{E}$$

There are more details to this calculation in the notes on computing field due to a line charge.

For most real-world situations, the electric field integral cannot be solved analytically. That is, you could most likely write down the integral, but it cannot be computed because there's no anti-derivative for the function that you would be trying to integrate. So we have to think of another approach – one that makes use of the principle of superposition, which we know the electric field obeys.