> What's the potential of the neutral line when it's tied to earth ground?
Actually it can be several volts.
Consider a few kW load at the end of the wire (very easy if you consider the kind of loads encountered in a building). The wire has a Resistance depending on it's material and cross-section and length.
If you connect your DVM between the Neutral Line and Earth at your wall socket with no load on that circuit (remembering it could be connected to several outlets), then you probably will get no voltage, but as soon as you start to switch on appliances you will begin to register a voltage. Now load up with a few kW (say an electric room heater) and report back with what you find...
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