Question

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

• A. $x=\frac{1}{2},y=\frac{1}{2}$
• B. $x=\frac{1}{2},z=\frac{1}{3}$
• C. $y=-\frac{3}{2},z=\frac{1}{2}$
• D. $y=\frac{1}{2},z=\frac{3}{2}$

Answer 1

The correct option is C $y=-\frac{3}{2},z=\frac{1}{2}$

$Length\propto \left[G{]}^x\right[C{]}^y[h{]}^z{M}^0{L}^1{T}^0\propto [M^{-1}{L}^3{T}^{-2}{]}^x\left[L{T}^{-1}\right]y[M{L}^2{T}^{-1}{]}^z{M}^0{L}^1{T}^0\propto [M{]}^{z-x}\left[L{]}^{3x+y+2z}\right[T{]}^{-2x-y-z}$
On comparing we get
$x=z=\frac{1}{2},y=\frac{-3}{2}$