Can 12v 3-12A battery power safely mega 2560? what is maximum input current for mega 2560 and UNO? Have a good day?
INPUT VOLTAGE (RECOMMENDED) is 7-12V for Arduino Mega 2560, It will be better you use a 12V DC-DC converter to connect your board to the battery.
Arduino boards will only draw the amount of currect they need to function from a supply unit.
If you want to use the 5V output of the Mega to power other things, be aware that the onboard voltage regulator can't deliver much current as the voltage difference multiplied by the current needs to be dissipated as heat and there is virtually no heat sink
Be careful using a 12 volt battery, batteries will often exceed the rated voltage when new and only supplying a small amount of current. A car battery may be almost 15 volts while being charged.
First, the current capacity of the source (battery) is irrelevant unless it is too small. Your AC line into your home can supply 15Amps to any given plug. If you plug a 7watt light into the socket the light only draws a small part of an amp.
The input voltage range of your board is 7V to 12V. 12V is at the upper limit and could cause over heating (and potentially failure) of your board if you power anything of significance from the boards 5V output.
I recommend you follow dozie's advice and get a power capable device to lower the voltage to 5V.
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