What happens when a battery is fully charged and is still connected to the power supply?
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Once the battery is fully charged, it will draw a small amount of energy. This is to maintain the state of charge. The charger will supply this energy and this mode is the Trickle Charging Mode.
Continual charging of the battery will gradually reduce its capacity over time, but this process will take several years. Once the charging is complete, with certain batteries like Li-ion, the energy will be wasted as heat (i^2*R, R=battery internal resistance). Thus, it will be wiser to unplug the charger when not in use. Thanks for asking.
It wont charge from then!
Batteries and power supplies are one and the same thing. A battery is a type of supply that diacharges and can be charge again. Whereas a charger doesnt have the facility to store charge. If a 12V battery is diacharged to 8 or say 9V , when you will connect it to a 12 V charger, it tends to charge itself and try to regain its power of 12V. It is obvious that the potential at two ends is unequal, hence a balance has to be established. But the Power supply is of fixed value and will remain constant. So it is the battery that has to cope up. Once it reaches 12V, the potential at both the ends i.e. the charger(which is already 12V) and the Battery(which before was 8-9V) will be same.