Question

If the radius of a current carrying conductor is halved how does the current change in the circuit?

Answer 1

Resistance of a conductor wire
   = R = Resistivity * Length/Area of cross-section

So if radius is halved, Area becomes1/4th and so resistance becomes 4 times.
So current becomes 1/4th, if the potential difference across the conductor remains same.

Answer 2

if the conductor has zero resistance then the current does not change 

if the resistance of the conductor is taken into account, then the resistance of the conductor will get 4 times as area becomes 1/4. 

and if resistance of the conductor is the only resistance then the current becomes 1/4 the of its original value