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I'm having a similar issue. It's strange. The tank was pretty much empty (had less than 1 gallon in it) for 20 years. The gauge read just slightly above the Empty hash mark. I pumped out that remaining gas and poured 3 gallons of fresh gas in. Turned the key and the needle went to slightly over 1/4, which is a bit inaccurate since 3 gallons is slightly LESS than a quarter tank. Here's the mystery: yesterday, I turned the key and the gauge went past Full. I bounced the back of the car so the gas in the tank would slosh around, turned the key again and the needle went back to slightly over 1/4. Added one gallon today and the SAME exact thing happened. Bounced the back end a few times and the needle went back to almost 1/2. What's up with THAT?? And 4 gallons is NOT a half tank. 8 gallons would be half...
is the needle supposed to go back to Empty when the key is shut off?
Not really.
As I've explained elsewhere, a gas gauge has 2 coils; one of which is always pulling the pointer toward Full whenever ignition is on, and one pulling toward Empty whose strength is determined by the sending unit. When power is turned off there's nothing pulling it anywhere, so it drifts to wherever it feels like.
The tach is somewhat similar; it has 2 coils that are set at 90° angles, such that one pulls the pointer toward 0 or directly opposite 0 on the scale (whatever value that happens to be) depending on whether it's energized in the positive or the negative direction; and one that pulls it 90° from 0 or directly opposite that, depending on the polarity of its applied voltage. Thus the tach can point to any point anywhere on the scale by the proper application of voltages. Similar to the gas gauge, all the pulling forces go away when power is removed, so the pointer just goes to wherever the spirit moves it to.