Question

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

Answer 1

Correct Answer - A
As force `F = (e^(2))/(4pi epsilon_(0)r^(2)) rArr (e^(2))/(4pi epsilon_(0)) = r^(2)F`
Putting dimensions of r and F, we get
`rArr [(e^(2))/(4pi epsilon_(0))] = [ML^(3)T^(-2)]` ...(i)
Also, force, `F = (Gm^(2))/(r^(2))`
`rArr [G] = ([MLT^(-2)][L^(2)])/([M^(2)])`
`rArr [G] = [M^(-1)L^(3)T^(-2)]` ..(i)
and `[(1)/(c^(2))] = (1)/([L^(2)T^(-2)]) = [L^(-2)T^(2)]` ..(iii)
Now, checking optionwise,
`=(1)/(c^(2)) ((Ge^(2))/(4pi epsilon_(0)))^(1//2)`
`= [L^(-2)T^(2)] [L^(6)T^(-4)]^(1//2) = [L]`