Is the wife divorced if the husband swears by the mentioned formula, and if the oath relates to knowing any other woman and not just committing adultery with her? And is there any expiation?

Your wife has no right to question you about whether you are in any forbidden relationship with another woman or to make you swear an oath about it. Since you swore an oath denying any relationship, intending sexual intercourse, no divorce occurs, because an oath is based on the intention of the one swearing it. You are not required to offer expiation for this oath.