The Young's modulus of steel wire of radius $r$ and length $L$ is $Y$. If the radius $r$ and length $L$ of the wire are doubled then the value of $Y$

Young's modulus $Y$ is an intrinsic property of the material of the wire.

It depends only on the nature of the material and its temperature, and is independent of the dimensions of the wire such as length $L$ and radius $r$.

Therefore, when the radius and length of the steel wire are doubled, its Young's modulus remains unchanged.

Answer: remains unchanged