Van DeGraaff Generator
There is a web site dedicated to the Van DeGraaff generator. Here are some of
the contents from that website.
SOURCE
Van de Graaff
Electrostatic Generator Page
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Other Websites: Van de Graaff Generator Plans & Instructions
Other Websites: Misc Van de Graaff Generators
Alternative HV: Marx and Cockroft-Walton
Other electrostatic projects here
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Some practical electrostatic calculations
It is often instructive to see a practical application of theoretical
principles, so here are some examples:
Maximum voltage on the Science First(tm) 400kV Van
deGraaff generator
One limit on the maximum voltage that a Van deGraaff generator can reach is
when the electric field strength at the surface of the sphere exceeds the
breakdown for the insulating gas it is immersed in. The section on Van deGraaff
generators describes other limitations such as arcing down the belt. For small
demonstration type generators operating in room air, however, the E field
limit is usually applicable.
Field at surface of a sphere = Voltage / Radius of Sphere
So, we use the handy approximation of 30 kV/cm for the breakdown strength of
air and rearrange the equation a bit to get:
MaxVoltage = 30 kV * Radius(cm)
The classic Franklin Lee designed Van deGraaff from Scientific American has
an upper electrode about 14 inches in diameter which actually has a minimum
radius of curvature of about 7 inches (it is a "squashed" sphere).
MaxVoltage = 30 kV/cm * 17.5 cm = 525 kV
The generator is advertised as reaching 400 kV, which is probably a realistic
assessment, given that it isn't perfectly smooth and that there is an insulating
column between the sphereoid and the ground plane at the base.
Capacitance and stored energy of 400 kV Van deGraaff
While the upper electrode on that Van deGraaff isn't perfectly spherical, we
can probably approximate it as one to calculate the capacitance and stored
energy. Note that here, we'll use the real radius of the sphereoid, as opposed
to the breakdown voltage calculation, where we used the smallest radius of
curvature.
r = 7 inches = .178 meters
Csphere = 4*pi*epsilon*r = 111.2 pF/meter * .178 meters = 19.8 pF
Now, let's calculate the stored energy: Is it a crack or a bang when we
discharge it?
Energy = 0.5 * C * V^2
Energy = 0.5 * 19.8E-12 * 4E5^2 = 1.5 Joules
(A handy form of the energy equation is: Energy (Joules)= 1/2 *
C(microfarads) * V(kilovolts) ^2. The units nicely cancel.)
Force from 400 kV Van deGraaff on a grounded sphere
Pretty much the first thing people do when they get a demonstration Van
deGraaff is to make sparks from the upper electrode. When I did this, I was
amazed at the electrostatic force on the grounded electrode I was discharging
to. It is one thing to intellectually know about Columb forces, and another to
feel them in your arm muscles (or to watch them move the electrodes around). As
it happens, calculating the field between two conducting spheres is a non
trivial task because the charge isn't evenly distributed. However, let's
approximate (since experiment will determine the exact number, we only need to
know if it is a tiny force or a huge one).
At 400 kV, the gap between the spheres will be a minimum of 10-15 cm. Let's
use 15 cm spacing of 15 cm radius spheres for the calculation. A first order
approximation will use Coulomb's law, and treat the sphere as a point charge at
the center. First question: How much charge?
q = CV = 19.8E-12 * 400E3 = 7.920E-6 Coulombs
I've approximated C as a sphere in free space, which certainly isn't the
case, but will do for an order of magnitude calculation
Now, let's use Coulomb's law
F = 1/(4*pi*epsilon) *q1 * q2 / r^2 = 1/111.2E-12 * 7.92E-6 * 7.92E-6 /
(.45^2)
= 2.78 Newtons (approximately)
About a half a pound, which is more than you would expect at first glance.
Have fun. That is the goal of many of the offerings here at ElectricianEducation.com.
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