While the alternator is my first suspect also you need to perform a more thorough test before deciding where the fault lies.
Alternator or battery?
You will need a Volt Meter to accurately perform these tests.
With engine off for 5 minutes. Check battery voltage:
12.6 V or higher = fully charged battery
12.4 V = battery 80% charged
12.2 V = battery 60% charged – May or may not have charge enough to crank engine.
12.1 V or lower Battery Severely discharged
Charge battery before proceeding if a voltage of 12.3 volts or lower is observed.
Charge a discharged battery for 10 hours using a 10 amp charger or
Charge a discharged battery for 3 – 4 hours using a 30 amp charger.
This will restore the battery to at least an 80% state of charge, it will take 36 to 48 hours at a trickle charge, after the fast charge is completed, to completely recharge a severely discharged battery in good condition, but 80% or better state of charge is sufficient to proceed with testing.
Let battery rest for 5 minutes and repeat voltage tests above. If battery reads below 12.4 Volts after being properly charged and resting 5 minutes it is bad – replace it.
With battery properly charged and passing voltage test above turn on the headlights for one minute, read voltage with headlights on, voltage should be above 12.0 volts, if it drops below 12.0 volts at the one minute mark the battery is bad.
If it passes the headlight load test shut the headlights off and the voltage should recover to 12.4 to 12.6 volts within 1 to 2 minutes for a healthy battery.
Start the engine and let it run for 1 to 2 minutes. Check the battery voltage with the engine running, it should be between 13.5 volts under Very Hot conditions 90 degrees (F) ambient temperature or higher, to 14.6 volts under Cool to Cold conditions 45 degrees (F)
or colder. Voltages between 14.0 and 14.4 are normal at all temperatures.
Voltage of 12.2 to 12.6 indicates alternator not charging. Check drive belt!
Voltage of 12.8 to 13.4 volts indicates either bad alternator (diode/rectifier) or bad voltage regulator or belt slippage.
Voltage above 14.6 volts indicates a bad voltage regulator. Voltages above 15 volts will result in rapid damage to the battery and will also cause frequent lamp failures.