Planck's constant h, speed of light c and gravitational constant G are used to from a unit of length L and a unit of mass M. Then the correct option (s) is / (are)

Options

A. `M prop sqrtc`
B. `M prop sqrtG`
C. `L prop sqrth`
D. `L prop sqrtG`

Answer

(a,c,d)

Explanation

Here, Dimnsional formula for (i) Length L = ?
(ii) planck's constant `h = ML^2 T^(-1)`
(iii) gravitational constant `G = M^(-1) L^3 T^(-2)`
(iv) speed of light `C = LT^(-1)` for length L
let L `prop h^x c^y G^z` or `L = kh^x c^y G^z`
`M^0 L^1 T^0 = k[ML^2 T^(-1)]^x (LT^(-1))^y [M^(-1) L^3 T^(-2)]^z`
`M^0 L^1 T^0 =kM^(x-z) L^(2x + y + 3z) T^(-x-y-2z)`
Equating the powers of M, L, T, we get
`x =z =0, x =z
2y + y +3z =1`
`-x -y -2z =0`
On solving, we get `x =(1)/(2), y= -(3)/(2), z = (1)/(2)`
`:. L = k h^(1//2) c^(-3//2) G^(-1//2)`
`L = k sqrt((hG)/(c^3)), L prop sqrth, L prop sqrtG`
option (c ) and (d) are correct.
Similarly, for mass M
Let `M prop h^x c^y G^z` or `M = kh^x c^y G^z`
or `M^1 L^0 T^0 = k[ML^2 T^(-1)]^x (LT^(-1))^y [M^(-1) L^3 T^(-2)]^z`
or `M^1 L^0 T^0 = kM^(x-z) L^(2x + y +3z) T^(-x-y-2z)`
Equating the power of M, L, T, we get
`x -z= 1 (i)`
`2x +y +3z =0` ... (ii) `-x-y-2z = 0` ..... (iii)
On solving , we get `x =(1)/(2), y = (1)/(2), z = -(1)/(2)`
`M =k h^(1//2) c^(1//2) G^(-1//2) = k sqrt((hc)/(G))`
`M prop sqrth, M prop sqrtc, M prop (1)/(sqrtG)`
Hence option (a) is correct.