Reference signal when measuring EEG When performing EEG, a number of electrodes are placed on the scalp to measure the electric potential difference between those locations and a reference. In the unipolar configuration, the reference is often the signal on another electrode (and is common to all measurement electrodes) or an average of the all measurement electrodes. In the bipolar configuration, the reference is specific to each measurement electrode (and is often another electrode located nearby). For examples, see here and here.

Questions:


For the unipolar configuration, why can't we just use Earth ground as our reference instead of having a reference electrode or performing an average of measurements on all the electrodes?
For the bipolar configuration, if the electrodes are too close to each other, wouldn't we measure very low voltages since much of the signal appearing on one electrode would also be picked up by the other?
For the unipolar configuration, why can't we just use Earth ground as
  our reference instead of having a reference electrode or performing an
  average of measurements on all the electrodes?


The ground has a different function than the reference. The reference is the electrode that the voltage is recorded against; a voltage is a potential difference so you need two electrodes. The ground is used for common mode rejection. The primary purpose of the ground is to prevent power line noise from interfering with the small biopotential signals of interest (source: Biopac). 


  For the bipolar configuration, if the electrodes are too close to each
  other, wouldn't we measure very low voltages since much of the signal
  appearing on one electrode would also be picked up by the other?


Yes. The smaller the distance, the better the rejection of unwanted signals. But with that, you're also risk to throw away the baby with the bath water.