Q154 / 155 · WBJEE 2025 · Physics

If the dimensions of length are expressed as $\mathrm{G}^{\times} \mathrm{C}^y \mathrm{~h}^{\mathrm{z}}$, where $\mathrm{G}, \mathrm{C}$ and h are the universal gravitational constant, speed of light and Planck's constant respectively, then

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

Solution

$\begin{array}{ll}\text { } L=\left[M^{-1} L^3 T^{-2}\right]^x\left[L T^{-1}\right]^y\left[M L^2 T^{-1}\right]^z \\ M^0 L T^0=M^{-x+z} L^{3 x+y+2 z} T^{-2 x-y-2} & \\ -x+z=0 ; \quad 3 x+y+2 z=1 ; \qquad 2 x+y+z=0 \\ x=z ; \quad \qquad 2 z=1 \qquad \qquad \qquad 3x+y=0 \\ \qquad \qquad \qquad z=1 / 2 \\ \qquad \qquad \qquad x=\frac{1}{2} \\ \qquad \qquad \qquad \text { and } y=-\frac{3}{2}\end{array}$

Hence, the correct answer is options: 'B', 'D'.

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