# Calculating the rate of electrostatic discharge via current density

After I asked this question, with the help of Floris' comment, I tried to learn more about the concept of current density and tried to calculate the current density in my scenerio which, it seems, is the only way I can determine the rate of discharge from a cylindrical object made out of dielectric material to an insulator. As stated in the Wikipedia page about current density,

A common approximation to the current density assumes the current simply is proportional to the electric field, as expressed by: $$\mathbf{J} = \sigma \mathbf{E} \$$ where E is the electric field and σ is the electrical conductivity.

However, I got stuck on what it means about the electric field here. For example, in my scenerio for a hollow cylinder with no top and bottom and with a charge $q$, radius $r$ and height $h$ that is made out of hard rubber whose electrical resistivity is $10^{13} \Omega /m$ at room conditions (25 C, 1 atm) do I have to calculate the magnitude of electric field applied on the surface of the hollow cylinder just beneath the insulating layer on top of the cylinder? In addition, for electrical conductivity do I have to determine the conductivity of cylinder or the conductivity of insulating layer? And can this method to calculate current density give me approximately accurate data or would I obtain a data that is way off from reality?

• @Cicero Oh, now I see. However, I still can't understand how I can use this to calculate current density since usually the application of Ohm's law is done for the same conductor but not two different materials in which one is enclosed by another as in my scenerio. – Starior Sep 12 '15 at 21:52
• I have to go somewhere so I couldn't finish my answer. I will try to answer tomorrow if i can. – Cicero Sep 12 '15 at 22:19
• @Cicero Ok, hope to see your answer then. Thanks for the effort. – Starior Sep 12 '15 at 22:20
• I decided to leave out the calculations and demonstrations and instead answer the conceptual questions in simple terms, leaving the calculation for you to enjoy. – Cicero Sep 14 '15 at 3:36

Note: I was going to give a fuller answer detailing the mathematical calculations and outlining a solution, using demonstrations from Resnick Halliday as well as Grifith, but I decided instead to give the answers you seeked and leave you the fun😀 of doing and confirming the actual calculations (so I left out the demonstrations and calculations).

This is an interesting question. I will answer the three parts separately.

Question 1: "what [does] it means about the electric field here … do I have to calculate the magnitude of electric field applied on the surface of the hollow cylinder just beneath the insulating layer on top of the cylinder?"

Generally current goes through a medium, for instance current going through a metal wire. In this case, the meaning of the electric field is unambiguous. Here however there is a current established between the cylinder and the surrounding insulating medium. However, the easiest way to solve this problem will be to do as Floris advised and consider the current flow inside the dielectric cylinder, which does indeed depend on the radius. Compute the electric field inside the medium and use the ohm's law approximation to determine the current density as a function of radius, and then integrate.

Question 2: "In addition, for electrical conductivity do I have to determine the conductivity of cylinder or the conductivity of insulating layer."

I would only consider current flow in the cylinder and thus use the conductivity of the cylinder, for we don't care about what happens when charge is discharged into the cylinder.

Question 3: "And can this method to calculate current density give me approximately accurate data or would I obtain a data that is way off from reality?"

This depends on whether the conditions by which the data was reflected were accurately reflected in the calculation process. That is, how the data was collected (the experiment) and how the calculation was performed (assumptions, models). You would have to specify both to determine the discrepancy. Here however are three possible reasons if there is a discrepancy:

0 -- experimental error (will assume to be negligible, which is why i said three rather than four)

1 -- the assumptions in the calculations were way off from reality

2 -- experiment operated in domains where the model doesn't work well (in high speeds in particle physics using Newtonian Mechanics will get bad results).

3 -- there were other processes occurring in the data collection which were neglected in calculations and turned out to be significant.

Happy calculating and experimenting!

• Thanks for the answer. I have one last question(if you haven't been tired of my questions yet 😀) Would the situation of charges staying in the cylinder while enclosed by an insulating layer be able to decrease the rate of charge loss in the system or would it be same if there was no insulating layer but all charges were present in a material with the same electrical resistivity as the insulating layer? – Starior Sep 14 '15 at 9:15