When I was little, I played around with batteries and wires and various electrical things that I’d taken to bits – enough to both burn my fingers (from shorting the battery) and give myself shocks (using a transformer to transiently step up my 9v battery to much higher voltages). Consequently, I was familiar with the word “voltage” and the vague hand-wavy description of it “being like water pressure” or “being like height”.
But then when I learned physics, I found that the primary focus was on the electric (really electrostatic) field – being a 3d vector field. Voltage aka potential difference made an appearance here, thanks to the nice property that for electrostatic fields you can think in terms of a scalar field (“the potential”), with the electric field being its gradient.
But it seemed like a big gulf between the “circuit theory viewpoint” and the “electrostatic viewpoint”. For example, if I take a lump of neutral material and grab some of its electrons and move them over a bit, I end up with a positively charged lump and a negative lump, and the surrounding E-field is a particular curvy shape (aka a “dipole field”). If I then think of this as a battery, and try to connect some wires to it, then my “circuit theory” viewpoint says that the electric field inside the wire must point along the wire the whole way – even if I tangle the wire into crazy knots. But how do we get from the nice dipole field to the twisty turny field required to follow the wire? It seems like two different worlds. Fortunately, there is an explanation – just not one that my Uni physics book mentioned. The best description of this comes from Matter and Interaction textbook (eg. here). When the wires are connected, there is an initial transient in which charges are displaced by the E-field, in such a way that a charge imbalance is created on the surfaces of the wires which causes the local E-field to reorient itself along the wire.
This is good news, because it means that you can indeed use all your electrostatic intuition of voltage and electric fields when dealing with circuits with batteries and resistors. The electric vector fields are all of the simple variety where you can succinctly summarise them using scalar fields. One key consequence of these ‘simple’ fields is that the work done by (or against) the electric field between two points is the same regardless of the route you take. This means we can talk about “the” voltage between point A and point B even when there’s multiple paths between them, because even though the E-field might be different along different paths, the net work done by (or against) the E-field along the entire path is always the same. Going one step further, if we agree to use a particular point (often labelled “ground”) as our reference point, we can even talk about “the voltage at point A” – but we understand that to mean “the potential difference between A and our chosen reference point”.
Voltmeters essentially rely on this “path independence” to do their job. In our nice electrostatic world, it doesn’t matter that we connect our voltmeter to points A and B using long winding leads, because even though the E-field might vary in space, we KNOW that the net work done by the E-field on a charge going through our voltmeter is the same as would be done by a charge going through the resistor (or whatever component we’re using the voltmeter to measure). Note: most measuring devices are named alluringly to suggest they directly measure the property you care about, when in practise they measure something else and use it to infer the value of interest. For example, an analog voltmeter actually measures the current through a known resistance, and we trust/hope that this is the same as the “volts” across whatever we’ve connected the leads to).
However, this lovely simple land of electrostatic fields with their path-independence isn’t the whole story. You can also create an electric field from a time-varying magnetic field (Faraday’s Law) and they’re of a very different nature – they form loops, quite unlike electrostatic fields which never form loops. With this kind of field, the work done going from point A to point B now DOES depend on exactly which path you choose. These fields cannot be represented as the gradient of a scalar potential. Consequently, we lose the lovely simple world in which we could think of point A as just having a single number (the potential) and point B having some other number/potential, and the work done by moving between then is just potentialB-potentialA – all of this is gone!
We do still have some useful properties – we can relate the work done around any closed path to the changing magnetic field (its flux) over a surface ‘capping’ that loop. But the notion of voltage as some path-independent value between point A and point B is lost. If we connect a “voltmeter” to points A and B, we’ll measure something pertaining to the path taken by the voltmeter’s leads and the voltmeter itself. But out ability to say “and that will be the same as any other path (such as through the component under test)” is gone. If we physically move the voltmeter and its leads in space, we will get a different reading (depending on the changing flux within the ‘loop’ consisting of the voltmeter and its leads and the circuit under test) – even though it is still connected to the same pointA and pointB in the circuit.
This paper is a great summary of what’s going on, and I was relieved to finally find it after many years to failing to reconcile my “physics” and “circuit” viewpoints! This other paper is also very good.
Incidentally, this makes it clear what Kirchhoff’s voltage law is coming from. It’s just a restatement of the properties of an electrostatic field – if you go from pointA to pointA in an electrostatic field, there’s no work done on a charge. But Kirchhoff came up with his rule long before anyone started thinking in terms of electric fields. It was also well before Faraday started futzing with induction and creating electric fields for which that property doesn’t hold. Essentially, Kirchhoff’s voltage law is a special case of Faraday’s law when there is no changing magnetic flux. If there is a changing magnetic field, the work done around a loop (the emf) is non-zero and Kirchhoff’s Law no longer holds.