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Qucik background info:

I am using a 117 Fluke, brand new bought in 2020 from Amazon.

enter image description here

What happened/happens:

This is a very simple circuit, but for some reason the resistor keeps heating when I try to use red LED's which are rated at 0.007 Amperes, so I tried to check the current going through the resistor, which caused the problem.

The LED simply goes out, and it won't turn on again. I tested this with a white LED also, same problem.

I can test the LEDs fine, and they show 0.0087 amps, which is fine, but the second I touch the resistor it burns up.

Does anyone have an idea, why this happens?

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    \$\begingroup\$ How did you insert the current meter in the circuit? Hopefully you didn't put it in current mode and then measure across the resistor. If you do that then you will short the resistor and cause a lot of current to go from the battery, through the meter and then to the LED. \$\endgroup\$ – user4574 Aug 7 '20 at 19:08
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    \$\begingroup\$ Also I noticed that you say the LEDs are rated at 7mA, but if you use a 100 ohm resistor and have a typical LED voltage around 2V then you are looking at nearly 70mA going through the part, which is 10X the rating. They may not burn up right away like that, but they won't last very long either. \$\endgroup\$ – user4574 Aug 7 '20 at 19:10
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    \$\begingroup\$ Yeah, an easier way is to measure the voltage across the resistor and divide by the resistance to find current. Your answer will be off by the tolerance of the resistor, but you'll avoid having to get in series with the circuit. \$\endgroup\$ – Stiddily Aug 7 '20 at 19:11
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    \$\begingroup\$ I think you are not getting your decimal points in the correct place. It is also more convenient and less prone to error if you use mA, not A. With 100-ohm resistor I would expect the current to be about 70mA (0.07A). The likely failure is as described by @user4574. \$\endgroup\$ – Kevin White Aug 7 '20 at 19:11
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    \$\begingroup\$ If you have solved your issue, don't edit the title to say "SOLVED"; instead, select the answer that helped you by ticking the check mark. This will also be reflected on the site's question lists as having been successfully answered. \$\endgroup\$ – JYelton Aug 7 '20 at 19:22
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You are not using the multi-meter correctly. To measure voltage you can just put the probes across the component you want to measure. It's like measuring water temperature. You just need to stick your finger in and feel it. No need to divert the water.

But measuring current is like measuring water flow. You need to divert all the water through your measurement device. When set to measure current, the multimeter probes will be changed they basically act as the same piece of wire since the multi-meter is expecting you to send all the current to be measured through it. If you place the probes across something in current mode, it diverts current and causes a short-circuit across the component.

You must actually break the circuit and stick your multimeter into it to measure current. When in ammeter mode, your two probes are basically the same piece of wire. If your meter is in current measurement mode and you stick the probes into your circuit across the resistor as if you were measuring voltage it will be as if you stuck a piece of wire across your resistor. This will divert all current that would otherwise flow through your resistor through the multi-meter instead. In other words, it shorts out your resistor and removes it from the circuit, then your LED explodes.

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You need to break the circuit and put the meter in SERIES with the LED+resistor+battery.

It will measure the same current wherever you put it, since it's a series circuit.

You are putting it in PARALLEL (across the LED) which means the LED is shorted (almost) and the current comes from the battery + resistor so you get (ideally) 90mA = 0.09A not 0.009A with 100 ohms (maybe you used a different value of resistor like 1K).

When you put it across the resistor, the current flows through the meter and LED with little resistance so the LED is damaged (can also blow the fuse in your meter). Some meters can be damaged by this (but the Fluke is a good product and is well protected, still it can be a pain to have to replace the fuse).

A cheap meter used this way across the mains can be physically dangerous. Some meters have an alarm beeper that goes off when the probe is inserted into the current measurement jack and the switch is set to a voltage range (it is easy to forget and stick the probes across a power supply, so it's a good habit to always return the probes to the voltage jacks).

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