Eddington's Electron Enigma

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The physicist Arthur Eddington once remarked that if 1 gram of electrons were to be confined to a sphere of radius 10cm, then the mass associated with their electrostatic potential energy would be between 10-20 million tons. We now proceed to investigate this claim.

This exercise investigates the strength of the electrostatic force that binds atoms together, and leads to suggestions that a different force (not gravity) is responsible for binding atomic nuclei.

Useful Data For These Groovy Sums


Q1) (a) (Requires integral calculus) If a charged sphere of radius 'r' is accreted in shells of thickness da and internal radius 'a', then show that in the limit if charge density per unit volume is s that the work done in accreting the whole sphere is given by the integral :
 
 



(b) Thus show that the electrostatic potential energy associated with the sphere is given by:

where 'q' is the total charge contained within the sphere.

Q2) (a) Calculate the number of electrons in 1g and hence calculate their total charge 'q'.

(b) Show that their potential energy is 1.67 x 1027 J.

(c) If E = mc2 then check whether Eddington's assertion is true.


Q3) (a) Now consider a sphere of water of 10cm radius. Calculate the mass of water in the sphere and hence calculate the number of moles of water present.

(b) Calculate the number of molecules of water present and by considering the electronic structure of water, show that the total number of electrons present in the water sample is approximately 1.4 x 10 27

(c) Hence calculate the mass of these electrons.


Q4) (a) Explain as fully as you can, the significance of these calculations.

(b) Explain why we are not faced with the situation that Eddington describes.


Q5) (a) Find out the radius of ,say, a gold nucleus.Calculate the mass associated with its potential energy. Compare this with its calculated mass in grams and comment.

(b) Using Newton's Gravitational Law and Coulomb's Law , calculate the ratio of the electrostatic to gravitational forces for the electron and proton in a hydrogen atom.

(c) Explain why a gold nucleus cannot be held together by gravity alone. What does this suggest about the forces that bind a nucleus together ?

USEFUL DATA
Mass of Electron = 9.1 x 10-31 Kg
Mass of proton = 1.67 x 10-27 Kg
Bohr Radius of Hydrogen atom = 5.3 x 10 -10 m
Universal Gravitational Constant = 6.67 x 10 -11 Nm2Kg-2
Electronic charge quantum = 1.6 x 10-19 C
Relative Molecular mass of water = 18g
Avogadro Number = 6.02 x 1023 mol-1
Speed of light in vacuo = 3 x108 ms-1
1/4pe0 = 9 x 109 Nm2C-1

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