I'll stick with the magnetized body. Steel vehicles are problematic for
compasses because they always introduce errors. Mostly the errors are not
big enough to cause trouble for a driver. Aircraft pilots have to know the
heading with precision but car drivers are happy with "kinda northeastish."
The problem is that when a car body is magnetized the compass is effectively
inside a magnet. That will make the compass point in one general
direction... in this case, ENE. How the car got that way and how to degauss
it, I surely don't know.
My guess is that the car body is magnetized. I used to work in aviation and
would periodically see the problem in fabric covered planes that had steel
tubing frames. We would try to compensate the compass and we wouldn't even
get in the ballpark. (If the problem is electrical, you will see the compass
jump when you turn things on and off.)
It is theoretically possible to degauss the car body - it's the practical
part that gets in the way. The essense is to wrap the body with wire (the
way you are probably already thinking, from the side over the top and down
and around). The wire is connected to AC power - you'd want a few amps
flowing - that is slowly decreased. I've never done it because it was beyond
the authorization of our avionics/instrument shop. Whether you want a
compass enough to pursue that is up to you.
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