How to Calculate Force for Van de Graaff Generator | 70-Year Old Challenge

[mattbeatlefreak]

Homework Statement

A fellow scientist heard that a Van de Graaff generator built 70 years ago could collect 5.0 C of charge on its dome, which had a radius of 1.1 m, and has challenged you to do the same. You plan to use the same dome with the same radius and the belt you plan to use is 200 mm wide and 10.0 m long (5.0 m to go up to the dome, and 5.0 m to come back down). Charging the belt gives it a surface charge density of 65 μC/m2 . Assume that the belt is being charged at a distance of half of the belt length from the center of the dome.

How much force must your motor be able to exert on the belt in order to accomplish your goal?

Homework Equations

F=qE
W=Fd=qV
ΔU=W

The Attempt at a Solution

Not sure how to tackle this problem. My thought is that to get a force, you could find the work done by the motor and divide by the distance. The work could be found by the change in electric potential energy to charge the generator. However, I don't feel confident with this approach.
I did calculate the surface area of the entire belt (2 m²) and the using the given surface charge density find the amount of charge over the entire belt (1.3*10^-4 C). That would mean that the belt would have to do a complete pass over 38,461 times...
I don't really have a clue on this problem, if anyone could say if I'm on the right track or help me get going, it would rock. Thanks!

 

[Bystander]

When's the motor working hardest? At the beginning of the process? Or, at the end?

 

[TSny]

Maybe treat the belt as a line charge as shown below. Can you find an expression for the force on a small section of the belt?

 

[mattbeatlefreak]

When's the motor working hardest? At the beginning of the process? Or, at the end?

Thanks for replying. I would think that as the charge is transferred onto the sphere, it would require more and more force to continue to apply more charge carriers. So then, at the end. Does this mean that I should only consider the amount of work it would take to move charge from the motor to the sphere? The belt is charged at a distance halfway up to the generator, so would it be the work it would take to move then 2.5 meters up to the sphere?

 

[Bystander]

Maybe treat the belt as a line charge

 

[andrevdh]

Shouldn't it be related to the voltage of the dome?

 

[TSny]

My approach was to concentrate entirely on the force without using energy or voltage. But, I think you can also get the answer using just energy concepts. First method will involve simple integration, second method won't require integration.

 

[mattbeatlefreak]

My approach was to concentrate entirely on the force without using energy or voltage. But, I think you can also get the answer using just energy concepts. First method will involve simple integration, second method won't require integration.

I solved using the integration method, and finally got the right answer. I was misreading the question slightly and my integration bounds were incorrect. Thanks for all the help everyone!