If Plank constant (h) speed of free space light (c) and Newton gravitation constant (G) are chosen as the fundamental quantities, then dimensional formula of length will be ......

If Plank constant (h) speed of free space light (c) and Newton gravita

1.	If Plank constant (h) speed of free space light (c) and Newton gravitation constant (G) are chosen as the fundamental quantities, then dimensional formula of length will be ......
Answer» `sqrt((hc)/(G))` `sqrt((G_(c))/(h^((3)/(2))))` `(sqrt(hG))/(C^((3)/(2))` `(sqrt(hG))/(c^((5)/(2)))` Solution :Length `lprop h^(a)G^(B) c^(d` where `a,b,d in R` `:. l=kh^(a)G^(b)c^(d)` `M^(0)L^(1)T^(0)=(M^(2)L^(2)T^(-1))^(a)(M^(-1)L^(3)T^(-2))^(b) (M^(0)L^(1)T^(-1))^(d)` `=M^(a-b)L^(2a+3b+d) T^(-a-2b-d)` `a-b=0 "" ...(1)` `2a+3b+d=1 "" ...(2)` `-a-2b-d=0 ""...(3)` `:.` From equation (2) and (3) `5b+d=1 "" ...(4)` and `-3b-d=0 "" ...(5)` Add by equation (4) and (5) `2b=1` `:.b=(1)/(2) "" :. a=(1)/(2)` From equation (3) `:.-d=a+2b=(1)/(2)+1=(3)/(2)` `:.d=-(3)/(2)` `l=h^((1)/(2))G^((1)/(2))c^(-(3)/(2))` `:.l=((hG)^((1)/(2)))/(c^((3)/(2)))` `= (sqrt(hG))/(c^((3)/(2)))`