I don't know how much your PC draws, but let's say 200 watts. Is the stereo on at the same time?
200 watts / 13 volts = 15 amps.
A typical car battery supplies perhaps 60 amp hours. This varies by type of battery, but it means that your battery can supply 60 amps for one hour before being discharged.
Batteries that are under higher load actually have less capacity. So, while you would think a battery could supply 120 amps for 1/2 hour, it would likely be less.
Depending on the amount of stuff you have on when your computer is running (ahem, a 1200 watt amp, for example), it is not surprising that you are down a volt in 1/2 an hour.
A capacitor will absolutely not help in this situation. The cap is charged from your battery and is designed only to momentarily supply the necessary voltage to the system when the stereo causes it to drop. No effect in longer term use.



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