Actually, you can have a current flow on a common pipe in certain fault conditions. For example, look at the case where two houses are fed from the same transformer and share a common conductive water pipe. If the neutral wire on the service drop is broken (on the street side the main bonding jumper between neutral and ground) on one of the houses, neutral current will flow through the common water pipe and back out the other house's neutral.
Is this the section where we answer homework questions or the NIST?
The OP did not specify a damaged ground, a fault conditon, or a high amperage that would cause an "earth potential rise".
He/she asked about the "earth wire" which is supposed to be connected to a buried pipe in the first place, and not carry any noticable current unless there is a fault condition.
i research something on how a 230V a.c. supply is derived from a 3-phase 415V transformer, noting also that a consumer's neutral wire connects to the transformer's star/ neutral point further away. A separate "protective" earth (the steel armour of the underground cable) runs between the consumer's earth terminal and the (earthed) star point of the transformer as well.
So:
It's widely understood that the need for "earthing" is a safety measure designed to prevent electric shock from wiring faults or insulation breakdowns.
I do not want to distract from the OPs question, so I won't. The real world is much more complex. If anyone wants to start a separate thread to discuss the exceptions start it and then PM me.
In an ideal world all grounds are the same. In the real world they aren't.
That exact thing happened at my house about a week or two ago. (I posted the problem and the cure on this site.) If the OP on this thread says anything to indicate a fault condition, I am ready to answer in minute detail.