My galaxy s3 was charging very slowly.
Micro usb port was very loose.
To Figure out the problem first; install Galaxy Charging Current Lite application to your galaxy S3 from google play store
Galaxy Charging Current Lite
After installing the application, Look at the Maximum field and hit refresh button. Unconnect/connect the charging cable many times and hit refresh button. Hold your charging cables micro usb port firmly to the phone.
Maximum value should be 999 or 1000 when charging the phone using AC adapter.
It should be around 400 when charging the phone from a computer's USB port.
If your maximum value is less, there is a problem. Try to find out the cause.
In my case, problem was the loose micro USB port. I used pliers to fix my problem.
Try it, may be your problem is the same as mine. Look at the following figure, and tighten your connector using pliers as I do. You just need to squeeze the middle part of micro usb port and it is gonna fix the loose connection issue.
Also, blow out to your phone micro usb socket to eliminate any dust.
If your problem is different, you can try buying a new AC charger and new micro usb cable. I hope the problem is not with your galaxy s3's socket or battery.
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