Potential Difference in a van der graaff generator

In summary, the conversation discusses a problem from the book 'Electromagnetism' by Grant & Phillips. The problem involves a cylindrical high-voltage terminal of a van der Graaff generator surrounded by an 'intershield' and a pressure vessel, with given radii. The goal is to determine the maximum potential difference that can be maintained between the terminal and the pressure vessel without causing breakdown in the gas inside. The given answers in the back of the book are 31.1 MV and 18.8 MV for the terminal and intershield, respectively. The conversation also includes a discussion of relevant equations and an attempt at solving the problem.
  • #1
JNBirDy
38
0
Hi, this problem is from 'Electromagnetism' by Grant & Phillips and it states,

Homework Statement


'http://i1129.photobucket.com/albums/m505/physicsbird1/126fig.png" [Broken] shows a cross-section of the cylindrical high-voltage terminal of a van der Graaff generator, surrounded by an 'intershield' and a pressure vessel, both of which are also cylindrical. The gas in the pressure vessel breaks down in electric fields greater than 1.6 x 10[itex]^{7}[/itex] volts/m. If the radii of the terminal, intershield and pressure vessel are 1.5 m, 2.5 m and 4 m respectively, what is the highest potential difference that can be maintained between the terminal and the pressure vessel?'

The answers given in the back of the book: '31.1 MV. When the terminal is at this voltage the intershield must be at 18.8MV to prevent breakdown.'


Homework Equations


d∅ = -E[itex]\bullet[/itex]dr

E = [itex]q/4πε₀r^{2}[/itex]

The Attempt at a Solution



1.6x10[itex]^{7}[/itex] = [itex]q/4πε₀(2.5)^{2}[/itex]

1.00x10[itex]^{8}[/itex] = [itex]q/4πε₀[/itex] (Let [itex]q/4πε₀[/itex] = A)

A = 1.00x10[itex]^{8}[/itex]


∅[itex]_{B}[/itex] - ∅[itex]_{A}[/itex] = -1.00x10[itex]^{8}[/itex][itex]\int^{4}_{1.5}[/itex](1[itex]/r^{2})[/itex]

= -1.00x10[itex]^{8}[-1/r]^{4}_{1.5}

= -1.00x10^{8} [-1/4 + 1/1.5]

= -4.17x10^{7}

So... can anybody tell me where I've went wrong?
 
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  • #3
You wrote, "E = q/4πε₀r^2"

E goes as 1/r ?
 

What is a van der Graaff generator?

A van der Graaff generator is a machine that generates high voltages of static electricity. It consists of a motorized belt that moves a rubber belt or metal sphere, which collects and stores electric charge. This stored charge creates a potential difference between the sphere and the base of the generator.

How does a van der Graaff generator work?

A van der Graaff generator works by separating electric charges using a motorized belt. As the belt moves, it rubs against a metal comb, creating a charge imbalance. The metal sphere or dome at the top of the generator collects and stores this charge, creating a potential difference between the sphere and the base of the generator.

What is potential difference in a van der Graaff generator?

Potential difference, also known as voltage, is the difference in electric potential between two points in a circuit. In a van der Graaff generator, the potential difference is the voltage created between the sphere and the base of the generator. This potential difference can reach very high levels, up to millions of volts.

What factors affect the potential difference in a van der Graaff generator?

The potential difference in a van der Graaff generator is affected by several factors, including the speed of the motorized belt, the material and size of the sphere, and the humidity of the surrounding air. The higher the speed of the belt and the larger the sphere, the higher the potential difference will be. Higher humidity can also lower the potential difference due to the buildup of moisture on the sphere.

What are some practical uses for a van der Graaff generator?

A van der Graaff generator has many practical uses, including in physics demonstrations, medical treatments, and industrial applications. It is commonly used in science classrooms to demonstrate static electricity and can also be used in particle accelerators to generate high-energy particles. In medicine, it can be used to treat conditions such as skin cancer. In industry, it can be used for coating materials with a thin layer of metal.

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