Question

A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics (c, e, mass of electron, mass of proton) and the gravitational constant G, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe (~15 billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?

Answer 1

The age of the universe is given by the relation,

t= ( e 2 4π ξ 0 ) 2 × 1 m p m e 2 c 3 G …… (1)

Here, t is the age of universe, e is the charge of electron, ξ 0 is the absolute permeability, m p is the mass of proton, m e is the mass of electron, c is the speed of the light and G is the universal gravitational constant.

It is known that

1 4π ξ 0 =9× 10 9   Nm 2 C -2

Substitute the values in expression (1).

t= [ ( 1.6× 10 −19 ) 2 ×9× 10 9 ] 2 × 1 1.67× 10 −19 × ( 9.1× 10 −31 ) 2 × ( 3× 10 8 ) 3 ×6.67× 10 11 =6× 10 9  years ≈6 billion years

So, the calculated age of the universe is 6 billion years. It is observed that the calculated age of the universe does not exactly coincide with the actual age of the universe, although it is very close.