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)

Options

A. `c^(2)[G(e^(2))/(4piepsilon_(0))]^(1//2)`
B. `(1)/(c^(2))[(e^(2))/(G4piepsilon_(0))]^(1//2)`
C. `(1)/(c^(2))G(e^(2))/(4piepsilon_(0))`
D. `(1)/(c^(2))[G(e^(2))/(4piepsilon_(0))]^(1//2)`

Answer

D

Explanation

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)`