I'm not sure how to ask this but,
I started learning about electromagnetism with how charged particles behave in electric fields. This was easy and intuitive to understand. But I felt a disconnect when moving on to circuitry, as voltage, something I think of as potential energy per coulomb, is used as the main variable in equations rather than electric field.
It's easy for me to understand why/how electrons move in a wire due to an electric field on one end (with a source of charged particles) and a ground on the other end of a wire. However, when that is replaced by voltage, it becomes less clear conceptually for me and makes voltage just feel like a magic number representing the 'pressure' in a wire.
This is a bit difficult to explain, but in short what I think I'm trying to say is, very roughly, why is it $I=V/R$, and not $I=E/R$ (I know units for $R$ would have to change). Would it not be better to treat the electric field as the main driving force and not voltage (at least conceptually)?
I'm sure I'm missing something important and would appreciate any help in understanding the issue. Apologies if the question is unclear.
Best Answer
For circuits the use of voltage is far better than the use of E fields would be.
First, and most importantly, the main benefit of circuit theory is that the geometry of a circuit is not part of the theory. When you are drawing and analyzing the circuit for a house it doesn’t matter if you run the wire under the floor or through a wall. The circuit is the same. That abstraction requires the use of voltage. The E field is a vector so it has magnitude and direction and hence the geometry is essential. Also, different resistor materials will have different E fields for the same resistance, depending on the length. So in using the E field the geometry of the components would be required. Most of the simplifications afforded by circuit theory would be gone, and circuits would essentially require Maxwell’s equations to solve instead of Kirchoff’s laws.
Second, common electrical sources are voltage sources, not E field sources. Consider two batteries of different sizes but using the same chemistry. Those two batteries will have the same voltage but different E fields. You can change the orientation of the electrodes inside the battery, or the spacing between the terminals, or any number of other design details. Those changes will change the E field, but not the voltage.
I am sure there are other reasons, but these two are strong. Especially the first. I don’t think that modern electronics would be what they are if we insisted on designing them using Maxwell’s equations instead of Kirchoff’s laws.