Nick, it's probably alternator noise, though any electrical motor (like your heater fan motor or fuel pump) can cause it too. Some radios are better shielded than others. I put a Cobra 18 in my 40, and no amount of filtering and etc. could get rid of the alternator noise, you could turn the squelch all the way up and it would still whine so bad you couldn't use the radio. On the advice of Clay's Radio Shop in Weatherford, I bought a Uniden 78 with an added noise filter inside the radio cabinet, and this has worked much better though when you turn the squelch down you can still hear a whine. 40's are very noisy electrically, compared to modern trucks.
If you're determined to try and make the Cobra 75 work, then you could try a few things. As has been said, wire it straight to the battery, both positive and negative. Put an in-line noise filter in the power line (can buy one at any truck stop). Make sure that the power wires do not run right next to the antenna coax. Put ferrite chokes over the antenna coax near both the radio end and the antenna end (can buy these from Radio Shack or a ham radio place). Add ground straps to your exhaust tailpipe, the antenna, and the alternator. All these things will help, though as in my case possibly not enough for a given radio.
