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I'm making the circuit attached, can anyone help me with what resistor value (marked by "?"). What value will need when I connect 2 UV LEDs into the circuit. The pack says LEDs are 3.1V.
Any help would be greatly appreciated...

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Check out my Instructable on Photo Components Testing.
https://www.instructables.com/id/Photo-Components-Testing/

Most UV LEDs run on 10ma so 9/.01=900 you should be good for 1 k ohm.

We need the forward current. If we say its 20mA, then the R, for a 9V supply is (9-(2x3.1)/0.02 = say, 290 Ohms.

According to my mathin'.... you could put 3 volts to each with a resistance of 60 Ohms. Being a strange value, you would probably need 6 10 Ohm resistors in series to achieve this. Hope this helps, and good luck!

What forward current are you assuming the LEDs have? Most UV LEDs i've seen have a forward current of 30mA. Your calculation seems to account for a forward current of about 50mA.

It all depends on what the forward voltage of the LEDs are. Look up one of the many LED calcs that are available online. Plug in how many LEDs you have, the source voltage, the forward voltage of the LEDs, and the forward current of the LEDs. Then it will give you the resistor value you'll need.