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I was thinking on this theoretical experiment for a while, but nothing useful came into my mind.

We have a "resistor" (rather a material like plastic) with a resistance expected to be very high, say 10^5 Mohms. Is it possible to measure the accurate vaule of the resistance with a max. 25V DC power supply (higher voltage damages the resistor) and a multimeter with max. 0.5μA resolution? (This is the main problem: if you connect the resistor and the meter serially to the scource, the meter won't show any current: \$ I = \frac{U}{R} = \frac{25}{10^{11}} = 2.5 \cdot 10^{-4}μA. \$ ) You can use any simple circuit element like resistors, capacitors, diodes etc., maybe really simple ICs like amplifiers at max. The solution does not have to be a circuit: the only aim is to get the accurate vaule of the resistance.

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    \$\begingroup\$ I'm crazy, so my first thought is to try putting 100kV across it and measure the current that way. :o) Assuming you have enough diodes and capacitors to make a voltage multiplier that large, and won't hurt yourself in the process. \$\endgroup\$ – Korozjin May 28 '15 at 20:57
  • \$\begingroup\$ @Korozjin Yes, it would be much easier if you could use higher voltages... but you cant, hence the power scource. I edited the task to clarify this anyway. \$\endgroup\$ – Neinstein May 28 '15 at 21:10
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    \$\begingroup\$ connect 1000 in parallel! \$\endgroup\$ – Chu May 28 '15 at 21:46
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You can do what you want, but it won't be simple. Scratch that, it will be fairly simple, but it won't be easy. As von Clausewitz said, "In war, the important things are very simple, and the simple things are very difficult." It applies here as well.

Actually, it's not as bad as it could be. Your nominal current is 250 pA. You can get op amps with bias currents in the 1 femtoamp range, so monitoring your current is certainly possible. I recommend you start with Bob Pease's "What's All This Femtoamp Stuff, Anyhow?" http://electronicdesign.com/test-amp-measurement/whats-all-femtoampere-stuff-anyhow followed by "What's All This Teflon Stuff, Anyhow?" http://electronicdesign.com/test-amp-measurement/whats-all-teflon-stuff-anyhow

The devil (or God, depending on your background) is in the details, and at these current levels details matter a whole bunch. You'll need to clean your apparatus with alcohol to get any fingerprints off, and grounding will take on a whole new meaning for you.

But it's certainly possible.

Basically, you're talking about making a picoammeter, and you can buy those commercially. Here, http://www.rbdinstruments.com/products/picoammeter.html?_kk=picoammeter&_kt=3ce3e0b6-02d3-4edb-8960-2273c533a323 for instance, is a cute little USB unit, but I'd be surprised if you can get it for less than $1500 - or maybe more. You can try eBay, and get rather cheaper than that.

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Existing equipment such as the Keithley Model 6517B Electrometer can measure resistances up to 10^16 ohms without using high voltages. This is possible because these instruments can measure extremely small currents, down to 10^-15 amperes or so.

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You can certainly get op-amps (as @Whatroughbeast mentioned) that can do this easily. Analog Devices have a few that have fempto amp bias currents and you'll need to arrange the op-amp as a transimpedance amplifier (TIA) with a feedback resistor of about 1Gohm. You can get this stuff fairly cheaply (less than $10). I have used such a set up for monitoring ion beams in a gas mass spectrometer.

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Do you have another resistor in the same range of known resistance? You could then use a wheatstone bridge to determine the resistance.

Actually the best way would be to have a number of resistors of known value, each half the resistance of the prior one. Then you could do a binary search.

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Check out this link: https://www.youtube.com/watch?v=xbKiAl2ioBg

I easily measure 200G ohm resistors using 10V. I have an accuracy of 0.1%. The curves in the video show the 0.1% line verses shot noise and Johnson noise. You need a low noise current measuring device near the shot noise levels.

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    \$\begingroup\$ It would be useful to provide a text description of the video. Adding as little as the name of the method you're using would make your answer searchable and readable by people who don't have youtube access. \$\endgroup\$ – Dmitry Grigoryev Jul 10 '15 at 11:52

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