Sections 15.1-15.2 in Matter and Interactions (4th edition)
In the previous notes, we talked about how to add fields using superposition, which can be greatly aided by the use of a computer (especially if there are many charges). We can also do a similar process for a line of charge analytically (using ideas from calculus) rather than using code. We will also talk about how these ideas ideas extend to large distributions of charge (including 2D sheets of charge and 3D volumes of charge). This page of notes will start by focusing on how we conceptually think of the electric field or electric potential for a line of charge (and we will go into the mathematical details in the next few pages).
Say we have a line of charge (e.g. a piece of tape), and we are interested in what the electric field looks like at some distance d away from the piece of the tape (Point A). How could we find the electric field at Point A? We cannot simply use →E=14πϵ0Qtotr2ˆr because the equation was built on the assumption that the charge was a point charge. However, we do know that the electric field follows superposition - meaning that electric field at a given point is the total (or sum) of all the electric fields from whatever sources are nearby.
So one way we could model the piece of tape would be to model it as two point charges - each point charge with half the total charge of the tape. In this case the electric field at Point A would be given by: →Etot=→E1+→E2. Here we have to calculate the electric field twice (find the different →r, find the magnitude of r, and calculate →E), but we get a better model. Now this might not be a great model for a line - but it's better than one point. We could make this model even better if we divided it into 4 point charges, spread out over the length of the tape, each with an amount of charge Q/4. Then the field would be given by →Etot=→E1+→E2+→E3+→E4. We have to do more work computationally, but we also get a much