If the energy, `E = G^p h^q c^r,` where G is the universal gravitational constant, h is the Planck's constant and c is the velocity of light, then the values of p are q and r are, respectively

Options

A. `-1//2, 1//2 and 5//2`
B. `1//2 , -1//2 and -5//2`
C. `-1//2, 1//2 and 3//2`
D. `1//2, 1//2 and -3//2`

Answer

A

Explanation

`E = G^(p) h^(q) C^(r )`
Dimensionally
`[M^(1) L^(2) T^(-2)] = [M^(-1) L^(3) T^(-2)]^(p) [M^(1) L^(2)T^(-1)]^(q) [LT^(-1)]^(r )`
`[M^(1) L^(2) T^(-2)] = [M^(-p) L^(3p) T^(-2p)] [ M^(q)L^(2q)T^(-q)][L^(r ) T^(-r)]`
`[M^(1) L^(2)T^(-2)] = [M^(-p+q) L ^(3p +2q+r) T ^(-2p-q-r) ]`
`rArr -p+ q =1 " "rArr 3p +2q +r=2`
`rArr -2p -q -r =-2`
Solving we get , `p = - (1)/(2), q= (1)/(2) , r = (5)/(2)`