The drift velocity of electrons in a metal is given by:
\(v=\frac1{enA}\)
Where,
I = current,
n = electronic concentration,
e = electronic charge,
A = area of the cross-section.
Now,
\(A=\frac{\pi d^2}4\)
When the diameter is halved, a new area of cross-section is
\(A'=\frac{\pi\left({\displaystyle\frac d{}}\right)^2}4=\frac14\frac{\pi d^2}4=\frac A4\)
Then new drift velocity is given as
\(v'=\frac1{en{\displaystyle\frac A4}}=4v\)
∴ v' = 4v, the drift velocity increases 4 times when the diameter of wire is halved.