#### 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?

#### Solution 1

One relation consists of some fundamental constants that give the age of the Universe by:

`t = (e^2)/(4piepsilon_0) xx 1/(m_p m_e ""^2c ""^3G)`

Where,

*t* = Age of Universe

e = Charge of electrons = 1.6 ×10^{–19} C

`epsilon_0` = Absolute permittivity

`m_p`= Mass of protons = 1.67 × 10^{–27} kg

`m_e`= Mass of electrons = 9.1 × 10^{–31} kg

c = Speed of light = 3 × 10^{8} m/s

G = Universal gravitational constant = 6.67 × 10^{11} Nm^{2} kg^{–2}

Also, `1/(4piepsilon_0) = 9xx10^(9) (Nm)^2"/"C^2`

Substituting these values in the equation, we get

t = `((1.6xx10^(-19))^4 xx (9xx10^9)^2)/((9.1xx10^(-31))^2xx1.67xx10^(-27)xx(3xx10^8)^3 xx 6.67xx10^(-11))`

=`((1.6)^4xx81)/(9.1xx1.67xx27xx.6.67)xx10^(-76+18+62+27-24+11)S`

=`((1.6)^4xx81)/(9.1xx1.67xx27xx.6.67xx365xx24xx3600)xx10^(-76+18+62+27-24+11) "year"`

`~~6xx10^(-9) xx 10^(18)` years

= 6 billion year

#### Solution 2

The values of different fundamental constants are given below:

Charge on an electron = `e = 1.6 xx 10^(-19)C`

Mass of an electron, `m_e = 9.1 xx 10^(-31) kg`

Mass of a proton `m_p = 1.67 xx 10^(-27) kg`

Speed of light, c = 3 xx 10^8 "m/s"

Gravitational constant, `G = 6.67 xx 10^(-11) N m^2 kg^(-2)`

`1/(4piepsilon_0) = 9 xx 10^9 Nm^2 C^(-2)`

We have to try to make permutations and combinations of the universal constants and see if there can be any such combination whose dimensions come out to be the dimensions of time. One such combination is:

`(e^2/(4piepsilon_0))^2. 1/(m_p m_e^2 c^3 G)`

According to coulomb's law of electrostatics

F = `1/(4piepsilon_0) ((e) (e))/r^2` or

`1/(4piepsilon_0) = (Fr^2) /e^2` or `(1/(4piepsilon_0))^2 = (F^2r^4)/e^4`

According to Netwon's law of gravitation,

`F = G (m_1m_2)/r^2 or G = (Fr^2)/(m_1m_2)`

Now , `[e^4/((4piepsilon_0)^2 m_p m_e^2c^3G)] = [e^4((F^2r^4)/e^4)1/(m_p m_e^2c^3) (m_1m_2)/(Fr^2)]`

=`[(Fr^2)/mc^3] = [(MLT^(-2)L^2)/(ML^3T^(-3))] = [T]`

Clearly the quantity under discussion has the dimensions of time.

Substituting values in the quantity under discussion we get

`((1.6xx10^(-19))^4(9xx10^9)^2)/((1.69xx10^(-27))(9.1xx10^(-31))^2(3xx10^(8))^3(6.667xx10^(-11)))`

=`2.1 xx 10^(16)` second

=`(2.1xx10^(16))/(60xx60xx24xx365.25)` year

=`6.65 xx 10^8` year

=10^{9} years

The estimated time is nearly one billion years.