How can I protect a picoammeter or a picoscope in HVDC measurements?
Can I connect a picoammeter to the ground electrode? If yes, please how do I protect the picoammeter so as not to damage it?


You can't measure amps without connecting an ammeter in series, so I'm not going to make a recommendation on this. If you want to measure current, get a current probe for your picoscope.


  Can I connect a voltage divider (with 5kohms and 50ohms connected in series) at the ground electrode, measure the voltage across the
  50ohms with a picoscope and from there I can use V=IR to get my
  current? If yes, please how do I protect the picoscope from getting
  damaged?


Yes you can use a voltage divider to lower the voltage to a device. If you used a 5k, that might be too low resistance. Lets say your using 200VDC, a 5k resistor will burn up ~8W so it had better be a big resistor rated for 10W or you could use a higher resistance value. 

One problem is that most of the picoscopes have a 200k input impedance, which is quite low for a scope, but that will cause a small amount of error depending on the resistors used for the voltage divider. 

One way to protect against transients is to use a zener diode, the picoscope can tolerate 50V (so it probably has 50V zener) just to be save you could put one or more zeners on the bridge between 20V and 50V to sink current if it gets outside of the picoscopes range.

You also didn't specify the input voltage or picoscope, so look up the specs for the scope you have. 







simulate this circuit &ndash; Schematic created using CircuitLab

250kV????

If you want to monitor high voltage like +100kV then your going to need a big resistor, and it will need to be protected from arcing. Why you ever thought you could do this with a 5k ohm resistor I'll never know (at 250kV that would be up to 12.5MW dissipated in the resistor, it would be toast, assuming that you have a large low impedance source) 

You'll need something like this, a 10000:1 scope probe, and even that will only get you to 200kV because the picoscope will clip at ±20V. I would recommend getting a scope that can handle a wider voltage range as it will probably be more expensive to get a 100000:1 probe if they even make them.