For those following along, this is how its supposed to work.
If its 12.5 volts at the power source, when the switch is turned on it should be 12.5 volts through the entire line to the bulbs.
In a perfect ground, it will be zero volts from the ground all the way back to the power source.
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The voltage difference across the fuse will be zero.
In the scenario below, there is a big voltage drop, when the switch is turned on. Its after the main splice because L1 and J1 work fine.
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Therefore most likely related to the shared fused.
Thats why we are measuring for voltage drop across the fuse.
Use the same technique to find resistance that is large enough to cause problems but too small to show up on a typical multimeter.
Test for drops along the flow path when current is flowing.
Change in Voltage = Current x Resistance.