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 ?

Explanation

Some constants of nuclear physics,
charge of electron `e=1.6xx10^(-19)C`
Light of speed in free space `c=3xx10^(8)ms^(-1)`
Universal Gravitation constant
`G=6.67xx10^(11)Nm^(2)kg^(-2)`
Mass of electron `m_(e)=9.1xx10^(-31)kg`
Mass of proton `m_(p)=1.67xx10^(-27)kg`
Permitivity in free space
`in_(0)=8.85xx10^(-12)Nm^(2)C^(-2)`
A physical quantity x is obtained of dimension of time by using these constants.
where `x=(e^(4))/(16pi^(2)in_(0)^(2)m_(p)m_(e)^(2)c^(3)G)`
By writing dimensions on both sides,
`[x]=([AT]^(4))/([M^(-1)L^(-3)T^(4)A^(2)][M.]^(2)xx[L^(1)T^(-1)]^(3)xx[M^(-1)L^(3)T^(-2)])`
`=M^(2-1-2+1)L^(6-3-3)T^(4-8+2+3)A^(4-4)`
`=M^(3-3)L^(6-6)T^(9-8)A^(0)`
`=M^(0)L^(0)T^(1)A^(0)`
`[x]=[T]`
Now, substituting values of constants in given equation,
`x=((1.6xx10^(-19))^(4))/(16xx(3.14)^(2)xx(8.854xx10^(-12))^(2)xx(1.67xx10^(-27))xx(9.1xx10^(-31))^(2)xx(3xx10^(8))^(3)xx(6.67xx10^(-11))`
`:.x=(6.5536xx10^(-76))/(34221942xx10^(-10))`
`=281xx10^(16)s`
`=6.9xx10^(8)` years
`=10^(9)` years
=1 Billion years
This calculation shows that value of constant is approximately equal to age of world.