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I have an rubber ESD anti static mat on my bench that came without the snaps. I installed 2 snaps, one on each side of the mat. I connected the ground wire from the snap on the mat to the ground in the wall outlet (the screw in the middle). The wire connecting to the ground in the wall has a 1mohm resistance and the wrist strap also has a 1mohm resistance.

I'm performing tests to see if the setup is working properly. Having my multimeter set to AC voltage, I connect the black lead to the ground on the mat, and I hold the other red lead in my hand. With the wrist strap off I read about 10V. When I have the wrist strap on, it reads 2V instead of 10V. I assume this means ground is working.

My question is why isn't the voltage dropping down to 0v instead of 2v when I have the wrist strap on?

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Your body is picking up AC voltage from some source. This is a common phenomenon.

The amount of current that is available is dependent on many factors. From the measurements that you give, I would approximate that current to be about 8V / (1 MegOhm + 1 MegOhm). Note that this doesn't include skin resistance but that is often quite low after you have been wearing a wrist strap for a while (sweat).

Note that the purpose of the mat is to dissipate high voltages caused static electricity. The resistor values that you mention will do this quite nicely.

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