Interestingly, most ammeters measure current by actually measuring the voltage drop across a very low precision resistance (so as not to load the CKT you are testing) and then doing a conversion (using Ohm's law). This is why it is easy to fry the ammeter section of your VOM if you aren't careful
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The voltmeter section of the VOM is a very high impedance CKT (also as not to load what you are testing) and thus uses negligible current. This is why you might see some noise on the meter if the box is a high impedance too...