A physical energy of the dimension of length that can be formula cut of `c,G` and `(e^(2))/(4 pi epsilon_(0))` is [`c` is velocity of light `G` is universal constant of gravilation e is change

Options

A. `(1)/(c^2)[(G e^2)/(4pi i"n"_o)]^(1//2)`
B. ` c^2[(Ge^2)/(4 pi "in"_0)]^(1//2)`
C. `(1)/(c^2)[(Ge^2)/(4pi "in"_0)]^(1//2)`
D. `(1)/(c ) [(Ge^2)/(4pi i"n"_0)]`

Answer

(a)

Explanation

`Let L prop c^x G^y ((e^2)/(4 pi "in"_0))^z`
`L = K c^x G^y((e^2)/(4 pi "in"_0))^z …. (i) `
where K = dimensionless constant. The dimensions of `c = LT^(-1)`
`G =(Fr^2)/(m_1 m_2) = [M^(-1) L^3 T^(-2)]`
`and (e^2)/(4pi i"n"_0) = Fr^2 = [ML^3 T^(-2)]`
From (i)
`M^0 L^1 T^0= (LT^(-1))^x [M^(-1) L^3 T^(-2)]^y [ML^3 T^(-2)]^z`
`M^(-y+z) L^(x + 3 y + 3z) T^(-x -2y -2z)`
Applying the principle of homogeneity of dimensions,
-y+z= 0 ....(ii)
x + 3y + 3z =1 ..... (iii)
-x -2y -2z = 0 ........(iv)
Add (iii) and (iv) : y+z = 1
Add (ii) `2z =1, z =(1)/(2)`
From (ii) ,`y= z = (1)/(2)`
From (iv), `-x =2y+ 2z = 2xx(1)/(2) + 2xx(1)/(2) = 2,`
x =-2
Put in (i), ` L = K c^(-2) G^(1//2)((e^2)/(4 pi "in"_0))^(1//2) = (1)/(c^2)((Ge^2)/(4pi i"n"_0))^(1//2)`
(omitting K)