Question

Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to:

• A. $\sqrt{\dfrac{h c^5}{G}}$
• B. $\sqrt{\dfrac{Gh}{c^3}}$
• C. $\sqrt{\dfrac{c^3}{Gh}}$
• D. $\sqrt{\dfrac{Gh}{c^5}}$

Hint:

The natural unit of time formed from $G$, $h$ (or $\hbar$), and $c$ is known as \textbf{Planck time}.

Answer 1

The Correct Option is D

Step 1: Write the dimensions of the given constants: \[ [G] = M^{-1}L^{3}T^{-2},\quad [h] = ML^{2}T^{-1},\quad [c] = LT^{-1} \]
Step 2: Assume the required time $T$ is proportional to: \[ T \propto G^{a} h^{b} c^{d} \]
Step 3: Substitute dimensions: \[ [T] = (M^{-1}L^{3}T^{-2})^{a}(ML^{2}T^{-1})^{b}(LT^{-1})^{d} \]
Step 4: Equate powers of $M$, $L$, and $T$: For mass $M$: \[ -a + b = 0 \Rightarrow b=a \] For length $L$: \[ 3a + 2b + d = 0 \] For time $T$: \[ -2a - b - d = 1 \]
Step 5: Substitute $b=a$ and solve: \[ 3a + 2a + d = 0 \Rightarrow d = -5a \] \[ -2a - a - (-5a) = 1 \Rightarrow 2a = 1 \Rightarrow a=\frac{1}{2} \]
Step 6: Hence, \[ T \propto G^{1/2} h^{1/2} c^{-5/2} = \sqrt{\frac{Gh}{c^5}} \]