Re: will a current flow in an object that is in an electric field?

First and most importantly, remember that electric charge cannot be created
or destroyed.  So, if a conducting object is placed into an external
electric field, charge can only move around on the surface in such a way
that the total charge is the same as before.  Remember also that the
electric field inside a conductor is zero (in equilibrium), and that this
means that the voltage on/in the conductor is uniform.  Any variations in
voltage from one part of the conductor to another would imply an electric
field, meaning that current would flow to cancel out the electric field.

So, to look at your question again, if a conductor were placed into an
electric field, would a current flow?  The answer is yes, but only
transiently.  This is true whether the conductor is grounded or "floating".

Just for the sake of argument, suppose that the conductor is a copper bar,
and that this bar is very very quickly placed into an electric field.  As
the bar is put into the electric field, the ends of the bar are, for just
an instant, at different voltages, just as you note.  However, this voltage
difference will immediately cause charge to flow within the copper to
cancel out the voltage difference: electrons will flow from the low-voltage
end of the bar to the high-voltage end.  As these electrons flow, the
resulting charge separation ("excess" electrons going to high-voltage end,
leaving a "deficit" of electrons at the low-voltage end) produces its own
electric field, which cancels out the external electric field.  Within a
very short time, the copper bar is in a new equilibrium state, with charge
distributed on its surface in a way which cancels out the external field,
leaving no electric field anywhere inside the bar.  With no electric field
inside, the entire bar will be at one voltage, which may or may not be
zero/ground.

Grounding the copper rod doesn't change the above answer, except to say
that some charge can flow from or to the rod, in order to keep its uniform
voltage equal to zero.  This "induced charge" on the rod (the charge that
came or went as a result of the ground connection) could, in principle, be
measured a couple of different ways.  One way might be to put a current
meter on the ground connection, and measure the current going into/out of
the rod.  Another might be to let the rod come to equilibrium, cut the
ground connection, remove the rod from the external electric field, and
then measure the electric field of the rod by itself (where the magnitude
of the electric field will be directly related to the amount of charge on
the copper rod).

Good luck!