Batteries are cheap. You will get your answer if you change the battery with a known good battery. iP4 battery replacement is very easy, but you must be careful because small fragile connectors are present. Also the bottom case screws are Pentalobe screws. Need Pentalobe screwdriver, or modified flat screwdriver or carefully used edge of square edge razor blade to remove them. Get the New Battery with a Pentalobe screwdriver. Also you will need a Phillips #00 screwdriver to remove the one battery connector screw.