Battery plugged in, not charging
This is not a defect or a problem. Your battery firmware or software driver is probably configured to forego charging when the battery is very near to 100%, to save on charge cycles. Rechargeable batteries have a limited lifespan, and repeated charging can shorten it. Charging the battery also heats it up, which can shorten its lifespan. The firmware is just trying to protect you from these and increase the lifespan of the battery.
If this situation is unfamiliar to you, then it is likely that you have used devices in the past which report false charge levels to the user. For example, some devices will say that they are "100%" charged, even though the battery's theoretical maximum capacity has not been completely filled up. This user interface trick is sometimes used to prevent consumers from being concerned by exactly the symptoms of a non-problem which you are seeing.
There is another possibility aside from an intentional limit to protect the battery. Laptops often (I can confirm for some Dell and HP models) have a specific signal from the charger, which informs the laptop of the charger's maximum capacity.
Sometimes, the wire that carries the signal or the corresponding part of the socket is damaged, and the laptop can no longer receive this signal. When this happens, a laptop might assume the charger is rated lower and refuse to charge the battery, ostensibly to protect the charger from overloading (which can cause a fire). Some will even throttle the CPU. Normally, the laptop will alert you both within Windows and at boot time with rather obvious messages - so, as long as you are not getting those messages, you should be fine.
If you are getting warning messages concerning the inability to detect the charger rating, the charger may have to be replaced, and/or the laptop may have to be repaired.
Some possible BIOS messages:
Power Adapter not recognized.
The AC power adapter type cannot be determined. Your system will operate slower and the battery will not charge. Please connect a Dell 90W AC adapter or higher for best system operation.
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