Question

Which one of the following combination of constants has dimensions of time?
[$G$ = constant of gravitation, $h$ = Planck's constant, $c$ = velocity of light]

• A. $\left[\dfrac{Gh}{c^{5}}\right]^{1/2}$
• B. $\left[\dfrac{Gh}{c}\right]^{1/2}$
• C. $\left[\dfrac{Gh}{c^{4}}\right]^{1/2}$
• D. $\left[\dfrac{Gh}{c^{3}}\right]^{1/2}$

Hint:

Always equate powers of $L$, $M$, and $T$ to identify the required physical dimension.

Answer 1

The Correct Option is A

Step 1: Write dimensions of constants.
\[ [G] = [M^{-1} L^{3} T^{-2}] \] \[ [h] = [M L^{2} T^{-1}] \] \[ [c] = [L T^{-1}] \]

Step 2: Find dimensions of $Gh$.
\[ [Gh] = [M^{-1} L^{3} T^{-2}] [M L^{2} T^{-1}] = [L^{5} T^{-3}] \]

Step 3: Divide by $c^{5$.}
\[ [c^{5}] = [L^{5} T^{-5}] \] \[ \left[\frac{Gh}{c^{5}}\right] = [T^{2}] \]

Step 4: Take square root.
\[ \left[\frac{Gh}{c^{5}}\right]^{1/2} = [T] \]

Step 5: Conclusion.
The given combination has the dimensions of time.