No, you don't have to put the license in each source code file.
If you look closer, most FOSS applications don't do that either. They put a copyright statement at the top of each file and a short sentence telling you what license the file is under and where you can find the full text of the license. They usually point you to the COPYING or LICENSE file containing the full text of the license and/or to a website that contains the full text (in case the COPYING file isn't there anymore).
Like Michael Borgwardt said in his answers, legally you don't have to do this. But it is advisable for source code that you intend to distribute since people can immediately see who has the copyright and what the license is.