Time required to build enough charge for an electric arc

[12emerta]

1.
A van der Graaf generator is used in classroom demonstrations to illustrate the production
of large electric fields with visible arcs. The threshold field for air to reach dielectric breakdown is
3*10^6 V/m.
In a particular demo, the van der Graaf dome is a hollow sphere of 25 cm radius that was discharged before the demo. At t=0, the charge-carrying belt is set to rotate at constant
speed so that 0.6*10^-6 C are put at the center of the dome every second. How long should it roughly take for an electric arc to form with an object held nearby?
(a) 15 s
(b) 35 s
(c) 50 s
(d) 120 s
(e) 250 s




2. 3*10^6=Voltage across gap/size of the gap



3. What I did was take 3 million and multiply by .25m. This gave me a value of 750,000 volts which is what is required for a spark to be made at that distance correct?
Now what i don't understand is how i would go from coulombs to volts with the given information

 

[haruspex]

For the potential near a charged sphere, there are plenty of online references, e.g. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potsph.html. However, I don't understand why you set the distance as 25cm. That's the radius of the sphere, not the distance to the object. The distance to the object would be useful if you knew the potential of the object. By dividing the potential of the sphere by the distance to the object you are implicitly assuming that the object is grounded.
Instead, find the strength of the field near the sphere. (http://farside.ph.utexas.edu/teaching/302l/lectures/node30.html)

 

[rude man]

What is the surface charge density necessary to generate an electric field of 3e6 V/m near the surface of the 25 cm radius sphere? Very simple formula.

 

[haruspex]

Btw, I believe the question ought to specify an object of zero charge, and either small or non-conducting. Otherwise it will alter the field.