I don't think I've seen a Birmingham Mint Royal Artillery badge. There's not one in the big 52 badge set they produced, but there does appear to be one in the much rarer Guards set. It's safe to assume that it will be an officers quality badge in gilt but with a chunky slider on the back.
Link:
http://www.britishbadgeforum.com/for...ctureid=121260
or:
Grey Green Acorns photo album.
Ian H