Let $Z_1, \ldots, Z_N \in \text{Gaussian}(0, 1)$ be independent standard gaussians. Using
R5444: Expectation of a squared standard gaussian random real number, we have
\begin{equation}
\mathbb{E} \chi
= \mathbb{E} \left( \sum_{n = 1}^N Z^2_n \right)
= \sum_{n = 1}^N \mathbb{E} Z^2_n
= \sum_{n = 1}^N 1
= N
\end{equation}
$\square$
