Question

The dimensions of length are expessed as $G^x{c}^y{h}^z$, where $G$, $c$ and $h$ are the universal gravitational constant, speed of light and Planck's constant respectively, then

• A. $x=\left(1/2\right),y=\left(1/2\right)$
• B. $x=\left(1/2\right),z=\left(1/2\right)$
• C. $y=(-3/2),z=\left(1/2\right)$
• D. $y=\left(1/2\right),z=\left(3/2\right)$

Answer 1

Answer: (B), (C)

The correct options are
B $x=\left(1/2\right),z=\left(1/2\right)$
C $y=(-3/2),z=\left(1/2\right)$

Let $L=G^x{c}^y{h}^z$
$\therefore \left[M^0{L}^1{T}^0\right]=\left[M^{-1}{L}^3{T}^{-2}{]}^x\right[L{T}^{-1}{]}^y[M{L}^2{T}^{-1}{]}^z$
or $\therefore \left[M^0{L}^1{T}^0\right]=M^{-x+z}{L}^{3x+y+2z}{T}^{-2x-y-z}$
$\therefore -x+z=0,3x+y+2z=1,-2x-y-z=0$

$\therefore x=\frac{1}{2},y=\frac{-3}{2},z=\frac{1}{2}$