What is the expectation of $R^2$ for for iid $y, x_1, ... x_k \sim N(0,1)$

$Y, X_1, ...., X_k$ are all iid $N(0,1)$ with $n$ samples.I can't make any progress on this problem... I don't even know an approximation but from simulation it seems to be $k/n$.

1. What is the expectation of $R^2$ for $ Y \sim \alpha + \beta X_1$?
2. What is the expectation of $R^2$ with $k$ independent variables?

For the case of one variable, I was trying to find the expectation of the correlation squared of $X_1,Y$ but I have no luck.

For the case of one variable, I was trying to find the expectation of the correlation squared of $X_1,Y$ but I have no luck.

I see this problem (Expected Value of R squared), but there doesn't seem to be a solution to the simpler gaussian case.