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...When a conductor is placed in contact with the terminals
of a battery, is the electric field constrained to exist only
within the conductor? ...
If the conductor is in contact with only one terminal, the situation
is static (no current flows). In such a static case, the field inside any
conductor is zero (all points in the conductor are at the same potential).
potential). The field between terminals in question 1 is then modified
by the presence of the conductor.