Which divisor for the sample variance?

Let $X_1, \dots, X_n$ be i.i.d. with mean $\mu$ and variance $\sigma^2$, and let $S = \sum_{i=1}^n (X_i - \bar{X})^2$.

(a) Compute $E[S]$. Which constant $c$ makes $S/c$ an unbiased estimator of $\sigma^2$?

(b) Now assume the data are normal. Which $c$ minimizes the mean squared error $E[(S/c - \sigma^2)^2]$?