Question about Van De Graaf generator

[Parad0x88]

Homework Statement

I have a sentence in my textbook that reads:

Because the "breakdown" electric field in air is about 3 X 10⁶ V/m, a sphere 1.00m in radius can be raised to a maximum potential of 3 X 10⁶ V

And now the problem states: The metallic sphere on top of a large van de Graaf generator has a radius of 2.0 m. Suppose that the sphere carries a charge of 3x10-5 C

Homework Equations

V = Q/4∏rε₀

The Attempt at a Solution

Does that mean to find the Volt, I have to simply multiply that 3 X 10⁶ V by 2, to give me 6 X 10⁶ V?

Or do I have to use the above equation to find the volt inside the generator?

 

[gneill]

Why not employ your Relevant Equation?

 

[Parad0x88]

Why not employ your Relevant Equation?

I guess it's the line from the text that confused me as to whether that equation was even necessary or not

 

[gneill]

The line in the text is telling you what limits the maximum potential on a surface (before breakdown and arcing occur to drain off the charge). If your calculated potential approaches this value, stand well back from the apparatus!

 

[Nickie]

I would like to clarify that the sentence in your textbook is referring to the maximum potential that can be achieved by a sphere with a radius of 1.00m. This is not the same as the potential achieved by a sphere with a radius of 2.0m. The maximum potential is dependent on the radius of the sphere and cannot simply be doubled.

To find the potential of the metallic sphere on top of the van de Graaf generator, we can use the formula V = Q/4πε0r, where V is the potential, Q is the charge, ε0 is the permittivity of free space, and r is the radius of the sphere.

Substituting the given values, we get V = (3x10^-5 C)/(4π(8.85x10^-12 C^2/Nm^2)(2.0m)) = 8.5x10^7 V.

Therefore, the potential of the metallic sphere on top of the van de Graaf generator is 8.5x10^7 V. This is significantly lower than the maximum potential of 3x10^6 V for a sphere with a radius of 1.00m. This is because the potential achieved by a sphere is inversely proportional to its radius, as seen in the formula V = Q/4πε0r.

I hope this helps clarify your confusion about finding the potential in this problem. It is always important to carefully read and understand the given information in a problem before attempting to solve it.