Is there a way to “prove” this with theory?
This is a pretty common real world practice of shutting the switch off and working on a light and leaving the rest of the circuit on in order to disturb less offices but I have heard of people getting hit on the neutral at the off light, they always say “it’s because there are other lights on the circuit” but I never understood this. Because to me electricity is lazy and is not going to back track to a room to go thru you to go to say building steel to get back. But I’d love to be able to “prove” it
For example, going by chapter 9 Table 9 of the NEC 3.31mm2, 12 AWG copper, has an AC resistance of 2 ohms per 1000 feet.
A 16 amp load will have resistance of a resistance of 17.3 ohms at 277 volts.
A 250 foot circuit will have a resistance of 0.5 ohms on the hot, and a resistance of 0.5 ohms on the neutral.
Adding 17.3 ohms of load, and 0.5 ohms on the hot we get 17.8 ohms from the circuit breaker panel to the neutral of the load at the far end. This forms part one of a resistive divider. The second part of the divider is the neutral going back to the panel which has a resistance of 0.5 ohms.
Using this calculator:
We get 7.5 volts drop across the neutral conductor between the panel and load. Those 7.5 volts will try find and take other routes back to the service if presented such as the human body.
Assuming body resistance of 1000 ohms we get 7.5 millamps of current which may be perceived. At 200 ohms body impedance (sweaty, grounded) we get 37 milliamps, which can not only be painful, but also could stop breathing worse case:
This alone is why the neutral should always be treated as live. Add to the fact that splices can open or go high resistance whereby the voltage drop of the neutral goes up.