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 varepsilon_0)]^(1/2)`
B. `c^2[G(e^2)/(4pi varepsilon_0)]^(1/2)`
C. `(1)/(c^2)[(e^2)/(G4pi varepsilon_0)]^(1/2)`
D. `(1)/(c )G(e^2)/(4pi varepsilon_0)`

Answer

A

Explanation

The dimensions of the quantities c, G and `(e^2)/(4pi varepsilon_0)` are as follows :
`[c]=[LT^(-1)]`
`[G]=[M^(-1)L^(3)T^(-2)]`
`[(e^2)/(4pi varepsilon_(0))]= [ML^(3)T^(-2)]`
Let the dimensions of the asked physical quantity (say) I be expressed as :
`[I]= c^(x)G^(y)((e^2)/(4pi varepsilon_(0)))^(z)`
Or, `[L]= ][L^(1)T^(-1)]^(x)[M^(-1)L^(3)T^(-2)]^(y)[M^(1)L^(3)T^(-2)]^(z)`
`-y+z=0 implies y=z " ""............"(i)`
`x+3y+3z=1" ""............"(ii)`
`-x-2y-2z=0" ""............"(iii)`
Solving equations (i), (ii) and (iii)
`x= -2, z=y=(1)/(2)`
Hence, I can be expressed as :
`(1)/(c^2)[G(e^2)/(4pi varepsilon_(0))]^(1/2)`