There usually wasn't any battery attached to a regiment. Artillery batteries were attached to brigades.

It appears the 10th often had a section or more of artillery attached to it, manned by soldiers of the 10th detailed for the purpose. This was probably because the 10th often acted in detached battalions and needed an organic artillery component. See the regimental history at http://www.rootsweb.com/~ilcivilw/history/c10cav.htm. The 10th had a battery of 4 small howitzers at the battle of Prairie Grove.