A physical quantity of the dimensions of length that can be formed out of c, G and e is [c is velocity of light, G is universal constant of gravitation and e is charge)

A physical quantity of the dimensions of length that can be formed out

1.	A physical quantity of the dimensions of length that can be formed out of c, G and e is [c is velocity of light, G is universal constant of gravitation and e is charge)
Answer» <P>`c^(2)[G(E^(2))/(4piepsilon_(0))]^(1//2)` `(1)/(c^(2))[(e^(2))/(G4piepsilon_(0))]^(1//2)` `(1)/(c^(2))G(e^(2))/(4piepsilon_(0))` `(1)/(c^(2))[G(e^(2))/(4piepsilon_(0))]^(1//2)` Solution :Dimensions of `(e^(2))/(4piepsilon_(0))= [F xx d^(2)]= [ML^(3)T^(-2)]` Dimensions of G=` [M^(-1)L^(3)T^(-2)]` Dimensions of `c= [LT^(-1)]` `l prop ((e^(2))/(4piepsilon_(0)))^(p) G^(q)c^(r)` `:. [L^(1)]= [ML^(3)T^(-2)]^(p)[M^(-1)L^(3)T^(-2)]^(q)[LT^(-1)]^(r)` On comparing both SIDES and solving ,we get `p= (1)/(2), q= (1)/(2)` and `r= -2` `:. [l]= (1)/(c^(2))[(Ge^(2))/(4piepsilon_(0))]^(1//2)`