How do you address a letter to a judge (not a court related matter, but not a friend-business associate).?

i read for court matters, you address the letter as Honorable Joe Brown and the salutation is Dear Judge Brown. Is this also correct for business letters?

The Honorable so and so... then Dear Judge Brown. That would be correct.

You would address the letter to Honorable Judge Smith...and refer to him/her as Your Honor, in the letter. If you are asking him/her something related to his/her position as a Judge.
If you are addressing them as a individual, I don't think it is necessary to use the formal address.
Hope this helps.

you always address them by their title no matter if it is personal or business matter unless they say that it is not necessary to do so.they have earned their title.

I would use his formal title ie the Honourable Judge Smith and your honor unless you knew him well.

I would address it Honourable Judge Joe Smith.
Give honour to whom honour is due . He earned it.And in talking to him address him as your honour.
If you are close friends that might be different because he might allow YOU to call him by his name in private but publicly you address him as your Honour. This is what I believe.

i would address this as judge brown etc etc