battery voltage question

I am having the same problem "sort of". The battery is located in the trunk and reads 13 1/2 the gauge under the dash is showing less than 11. Now I have the positive from the voltage gauge hooked to the ignition coil. ( I didn't know where else to run it.) I had the alternator tested a while back and they said that it was weak. Could this be causing the voltage drop and could this be the reason that the battery is showing weak at idle? I do have a ground cable going from the engine bock to the frame and the other ground is hooked to the rear fender well (for now). Lemme know what you think. Thanks in advance.