Your file will probably be cached - but it depends...
Different browsers have slightly different behaviors - most noticeably when dealing with ambiguous/limited caching headers emanating from the server. If you send a clear signal, the browsers obey, virtually all of the time.
You'll find that by far greatest variance is in the default caching configuration of different web servers and application servers.
Some (e.g. Apache) are likely to serve known static file types with HTTP headers encouraging the browser to cache them, while other servers may send no-cache
commands with every response - regardless of filetype.
...
So, first off, read some of the excellent HTTP caching tutorials out there. (HTTP Caching & Cache-Busting
for Content Publishers was a real eye opener for me :-)
Next install and fiddle around with Firebug and the Live HTTP Headers add-on , to find out which headers your server is actually sending.
Then read your web server docs to find out how to tweak them to perfection (or talk your sysadmin into doing it for you).
...
As to what happens when the browser is restarted, it depends on the browser and the user configuration.
As a rule of thumb, expect the browser to be more likely to check in with the server after each restart, to see if anything has changed (see If-Last-Modified and If-None-Match).
If you configure your server correctly, it should be able to return a super-short 304 Not Modified (costing very little bandwidth) and after that the browser will use the cache as normal.