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Macbook Air 2013, 13" which is not charging the battery but it shows on the battery icon that it is charging the battery (when in fact it does not). What are the possible things that can cause this and which is the most likely/common one? I would appriciate some answers to my question.

Lets start off by checking the state of the battery and it’s charging. Download this great app: CoconutBattery post a screenshot of the main window here so we can see it too. In addition, lets inspect the MagSafe charger to make sure its the correct one and the connectors are in good shape. Review this Apple T/N: Apple Portables: Troubleshooting MagSafe adapters