Question

If the dimensions of length are expressed as a $G^x\cdot C^y\cdot h^z$, where G, C and h are the universal gravitational constant, speed of light and plank constant respectively, then value of x,y,z will be :

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

Answer 1

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

Length $\propto G^x{C}^y{h}^z$
$\left[L\right]=\left[M^{-1}{L}^3{T}^{-2}{]}^x\right[L{T}^{-1}{]}^y[M{L}^2{T}^{-1}{]}^z$
On comparing the power of M, L and T in both sides, we get
$-x+z=0$ ....(i)
$3x+y+2z=1$ ....(ii)
and $-2x-y-z=0$ ....(iii)
By solving Eqs. (i), (ii) and (iii) we get
$x=\frac{1}{2};y=-\frac{3}{2}$ and $z=\frac{1}{2}$