What are the effects of a Dispel Magic on a Conjure Elemental spell?

If you cast a “successful” Dispel Magic in the following cases, what are the possible outcomes?

  1. An active Conjure Elemental spell, i.e. the caster is maintaining concentration at the time you cast Dispel Magic on it.
  2. An elemental that is the result of a Conjure Elemental spell where concentration has been lost, i.e. when the following condition described in the spell has occurred:

If your concentration is broken, the elemental doesn’t disappear. Instead, you lose control of the elemental, it becomes hostile and might attack. It can’t be dismissed by you, and disappears 1 hour after you summoned it. (PHB p.225)

I would like to know in what case, if any, the Elemental disappears as a result of the Dispel Magic.