I need to know who my public defender is

Are you in the US? Are you out on bail? A PD will contact you. Make sure your contact information is good. You could call the Court to ask how long it takes, in general, but in crowded areas, it will vary, and people will tell you to just wait.
PDs are horribly overworked. Many will see you for just a few minutes, if they assume that you are guilty and are going to cop a plea, with details about priors and so on taken into account, maybe ask for community service, etc.
Law done like a fast food restaurant. Cases go by so fast that you often can't even hear the rapid mumbling of both judge and lawyer.
We can't know. Too many details.

You find out after the court assigns one to you. Which is after means testing to show you can't afford a regular lawyer. Meaning you don't get one if you have the assets to hire your own. Its wrong to automatically assume you are entitled to one and will get one. ( in the USA)

You won't know until they contact you after an assignment is made.

Have you been charged with a crime in court? Were you arraigned? A PD is not assigned until that point.