A great physicist of this century (P. A. M. Dirac) loved playing with numerical values of fundamental constant of nature. This led him to an instreasing observaion. Dirac found that form the basic constant of atomin physice (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). Form the table of fundamental constants in this book, try to see if you too can construct this number (or any other instresting number you can think of). if its coincidence with the age of the universe ware significant, what would this imply for the constancy of fundamental constants ?

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Answer Text

Solution :
The values of different fundamental constants are given below : <br> `{:("Charge on an electron,",e=1.6xx10^(-19) C),("Mass of an electron,",m_(e)=9.1xx10^(-31) kg),("Mass of a proton",m_(p)=1.67xx10^(-27) kg),("Speed of light",c=3xx10^(8)m//s),("Gravitational constant,",G=6.67xx10^(-11) N m^(2) kg^(-2)):}` <br> `1/(4pi epsi_(0))=9xx10^(9) Nm^(2) C^(-2)` <br> We have to try to make permutations and combination 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 : <br> `(e^(2)/(4 pi epsi_(0)))^(2). (1)/(m_(p) m_(e)^(2)c^(3)G)` <br> According to Coulomb's law of electrostatics, <br> `F=1/(4pi epsi_(0))=((e)(e))/r^(2)` <br> or, `1/(4pi epsi_(0))=(F r^(2))/e^(2)` or `(1/(4 pi epsi_(0)))^(2)=(F^(2)r^(4))/e^(4)` <br> According to Newton's law of gravitation, <br> `F=G (m_(1)m_(2))/r^(2)` or `G=(Fr^(2))/(m_(1)m_(2))` <br> Now, `[e^(4)/((4pi epsi_(0))^(2) m_(p) m_(e)^(2)c^(3)G)]=[e^(4)((F^(2) r^(4))/e^(4))1/(m_(p)m_(e)^(2)c^(3))(m_(1)m_(2))/(Fr^(2))]` <br> `=[(Fr^(2))/(mc^(3))]=[(MLT^(-2)L^(2))/(ML^(3)T^(-3))]=[T]` <br> Clearly, the quantity under discussion has the dimensions of time. Substituting values in the quantity under discussion, we get <br> `((1.6xx10^(-19))^(4)(9xx10^(9))^(2))/((1.69xx10^(-27))(9.1xx10^(-31))^(2)(3xx10^(8))(6.67xx10^(-11)))` <br> `=2.1xx10^(16)` second <br> `= (2.1 xx10^(16))/(60xx60xx24xx365.25)` years <br> `=6.65xx10^(8)` years <br> `=10^(9)` years <br> The estimated time is nearly one billion years.

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