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, q and r are respectively:
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

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 )`
Using dimensions we can write,
`[ML^(2)T^(-1)]=[M^(-1)L^(3)T^(-2)]^(p)[ML^(2)T^(-1)]^(q)[LT^(-1)]^(r )`
`therefore [M^(1)L^(2)T^(-2)]=[M^(-p)L^(3p)T^(-2p)][M^(q)L^(2q)T^(-q)][L'T^(-r )]`
`=M^(-p+q)L^(3p_r)T^(-2p-q-r)`
Comparing the powers of M, L and T we get
`therefore -p+q=1" ...(2)"`
`3p+2q+r=2" ...(3)"`
`-2p-q-r=-2`
`therefore 2p+q+r=2" ...(4)"`
Solving (3) and (4) we get
`{:(" 3"p+2q+r=2),(" 2"p+q+r=2),(" (-) (-) (-) (-)"),(bar(" "p+q=0" ")" ...(5)"),(" "-p+q=1" ...(2)"),(bar(" "2q=1" ")):}`
`q=(1)/(2)`
From (5) `therefore p=-q=-(1)/(2)`
`therefore p=-(1)/(2)`
From (4) `therefore r=2-2p-q`
`=2-2(-(1)/(2))-(1)/(2)`
`=2+1-(1)/(2)=2(1)/(2)=(5)/(2)`
`r=(5)/(2)`
`therefore p=-(1)/(2), q=(1)/(2) and r=(5)/(2)`