Question

If dimensions of length are expressed as $G^x{C}^y{h}^z$ where $G$, $C$ and $h$ are universal gravitational constant, speed of light and Planck's constant respectively. Then

• A. $x=y$
• B. $x=z$
• C. $y=-3/2$
• D. $z=1/2$

Answer 1

Answer: (B), (C), (D)

$x=z$
$z=1/2$
$y=-3/2$

$\left(M^0{L}^1{T}^0\right)$ $\alpha G^x{l}^y{h}^z$

$G=\frac{\left(M^1{L}^1{T}^{-2}\right)}{M^2}\times \left(L^2\right)=M^{-1}{L}^3{T}^{-2}$

$C=L{T}^{-1}$

$h=\frac{Energy}{\frac{C}{\lambda }}=\frac{M^1{L}^2{T}^{-2}}{\frac{L{T}^{-1}}{L}}=M^1{L}^2{T}^{-1}$

$M^0{L}^1{T}^0=\left(M^{-1}{L}^3{T}^{-2}{)}^x\right(L{T}^{-1}{)}^y(M^1{L}^2{T}^{-1}{)}^z$

$-x+z=0,3x+y+2z=1,-2x-y-z=0$

$x=z$

$\Rightarrow 5x+y=1$

$y=-3/2$

$x=1/2$

$z=1/2$