Something wrong in the charging system. There are 3 parts to any charging system, the alternator, the regulator that controls the flow of electric charge to battery, and of course the battery. Determine if the humm comes from under the hood or from inside. If inside and related to the charging problem, it would be one of the computer control parts on american cars, but on Toyota it seems they've have kept the regulator as a separate component.

http://www.autozone.com/batteries-starting-and-charging/voltage-regulator/toyota/corolla

To localize where the hum comes from, use a paper towel tube, or make a paper tube, put by your ear and aim at suspected locations till you find it.

Realize the hum may not be associated with the charging system problem at all.