#### Answer

A force of 4000 N is needed to stretch the wire by 1 mm.

#### Work Step by Step

We can use Young's modulus to solve this question:
$Y = \frac{F/A}{\Delta~L/L}$
We can find an expression for the force required to stretch a wire;
$Y = \frac{F/A}{\Delta~L/L}$
$F = \frac{Y~A~\Delta L}{L}$
$F = \frac{Y~\pi~R^2~\Delta L}{L}$
We can write an expression for $F_1$;
$F_1 = \frac{Y~\pi~R_1^2~\Delta L}{L_1}$
We can write an expression for $F_2$;
$F_2 = \frac{Y~\pi~(2R_1)^2~\Delta L}{2L_1}$
$F_2 = 2\times\frac{Y~\pi~R_1^2~\Delta L}{L_1}$
$F_2 = 2~F_1$
$F_2 = (2)(2000~N)$
$F_2 = 4000~N$
A force of 4000 N is needed to stretch the wire by 1 mm.