For a circular loop of radius $R$ carrying current $I$: $B_{\text{center}} = \mu_0 I/(2R)$. For $N$ turns: $B = \mu_0 NI/(2R)$. Direction: right-hand rule — curl fingers along current, thumb points in field direction (along the axis). For a semicircular arc: $B = \mu_0 I/(4R)$. For an arc of angle $\theta$ (in radians): $B = \mu_0 I\theta/(4\pi R)$. Straight wire segments passing through the center contribute zero field (since $\vec{dl} \parallel \hat{r}$, making $\vec{dl} \times \hat{r} = 0$).