You lost neutral! This is an outage and an emergency
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Your neutral wire no longer connects back to the utility. Out of desperation, neutral is pathing through your *neutral-ground equipotential bond*, the ground rod and the dirt... to somebody else's ground rod and *their* N-G bond and neutral. Dirt is a terrible conductor, so this results in crazy voltages.

USUALLY this is the power company's service drop to your house.  That is two insulated aluminum wires hung off a bare aluminum carrier wire, which is anchored at both ends. This is both neutral and carrier, and if whips in the wind.  Aluminum has no fatigue limit. So the wire snaps! 

Often you can walk outside and look at the power company service drop wires and see if the carrier wire appears to be poorly attached or broken. Check both ends. 

**This wire is the power company's responsibility, and they will repair it fast for free, since it is a legitimate power outage**. 
 

What is happening is that without a neutral, nothing keeps 120V at 120V anymore.  Now your hot wires are still 240V apart, but your two banks of 120V are drifting all over the place (yet adding up to 240V).  This can be hard to detect, but it can also fry your appliances. 

If this is at an outbuilding with a subpanel, check your house; if wired old-school (3-wire), then the problem is the subpanel feeder. 

You can test this by checking voltages all over the house.  A lost neutral is indicated by two groups of near voltages, one group *under* 120V and the other *over* by same amount but totaling to 240V-ish (e.g. 105 and 137). If you power up a high-power 120V appliance, this voltage difference changes (e.g. 81 and 160). 


**Footnote:** It is *possible* that the current on your grounding electrode isn't caused by your own lost neutral, but rather, your neighbor's. However, I doubt that because you switching on large 120V loads has direct effect on the symptom. That wouldn't be so if your neutral was OK.