Question

If the dimensions of length are expressed as \( G^x c^y \hbar^z \), where \( G \), \( c \) and \( \hbar \) are gravitational constant, speed of light and Planck’s constant respectively, then

• A. \( x=\frac12,\; y=\frac12 \)
• B. \( x=\frac12,\; z=\frac12 \)
• C. \( y=\frac12,\; z=\frac32 \)
• D. \( y=\frac32,\; z=\frac12 \)

Hint:

Planck scale: \begin{itemize} \item Planck length uses \( \sqrt{\frac{G\hbar}{c^3}} \). \end{itemize}

Answer 1

The Correct Option is B

Concept: Dimensional analysis Dimensions: \[ [G] = \frac{L^3}{MT^2}, \quad [c] = LT^{-1}, \quad [\hbar] = ML^2T^{-1} \] Step 1: {\color{red}Write dimensional equation.} \[ L = G^x c^y \hbar^z \] Step 2: {\color{red}Substitute dimensions.} \[ L = (L^3 M^{-1} T^{-2})^x (L T^{-1})^y (M L^2 T^{-1})^z \] Step 3: {\color{red}Equate powers.} Mass: \[ -x + z = 0 \Rightarrow z=x \] Time: \[ -2x - y - z = 0 \] Length: \[ 3x + y + 2z = 1 \] Step 4: {\color{red}Solve.} Using \( z=x \): Time: \[ -3x - y = 0 \Rightarrow y=-3x \] Length: \[ 3x -3x + 2x = 1 \Rightarrow x=\frac12 \] \[ z=\frac12 \]