Electric potential across a cross sectional cylinder

[Londonfish]

Homework Statement

A cross-section of a cylindrical high voltage terminal (inner cylinder) of a van de Graaff generator, surrounded by an 'intershield' (middle cylinder) and a pressure vessel (outer cylinder). The gas in the pressure vessel breaks down in electric fields greater than 1.6*10⁷volts/m. If the radii of the terminal, intershield and pressure vessal are 1.5m, 2.5m, 4m respectively, what is the highest potential difference that can be maintained between the terminal and the pressure vessel? (hint: the intershield must be maintained at a potential such that breakdown is about to occur on its own outer surface as well on the surface of the terminal.)

Homework Equations

r_terminal=1.5m
r_intershield=2.5m
r_{pressure vessel}=4m
E_{max, p-vessel}= 1.6*10⁷volts/m

∫E*ds= q/ε (Gauss's Law)
I believe E= q/ (2*pi*r*ε);where r= radius

V(r_A) - V(r_B) = -∫ q/ (2*pi*r*ε) dr ; V= potential energy, integrate from A to B

The Attempt at a Solution

E_max,p-vessel= q/ (2*pi*r*ε)
q=E_max*2*pi*ε*(4-2.5)
q=0.001335 C.
Would this q work for the entire cylinder if I changed the radius or is it just for the pressure vessel because that's where the electric field is?

Then,
V(r_A) - V(r_B) = -∫ q/ (2*pi*r*ε) dr
= -q/ (2*pi*ε)∫1/r dr
= -q/ (2*pi*ε)(ln(B)-ln)A) ; Where A,B would be the radius
plugging in q and using r_i=2.5m and r_pv=4m
= (-2.4*10⁷)*(-0.47)
= 1.128*10⁷ volts or 11.28 megavolts

I don't really know where to go from here, or even if this step is true. Is there any easier way I could be doing this?

 

[Spinnor]

JNBirDy is doing the same problem, he has the answers, do you? See,

https://www.physicsforums.com/showthread.php?t=536751

 

[Spinnor]

Are you treating this as two problems, first ignore the inner cylinder. You know the maximum electric field at the surface of the shield cylinder. As the electric field goes as 1/r you can work backwards and determine the charge density and thus the electric field as a function of r, with that you can determine the potential difference between the shield and the outer cylinder.

Now you do the same for the inner cylinder, you will get a potential difference between the inner and shield cylinders? The sum of those potential differences is the potential difference between the inner and outer cylinders?