Which one of the following combination of constants has dimensions of time? $[\mathrm{G}=$ constant of…
Which one of the following combination of constants has dimensions of time? $[\mathrm{G}=$ constant of gravitation, $\mathrm{h}=$ Planck's constant, $\mathrm{c}=$ velocity of light $]$
$\left[\frac{G h}{c^{5}}\right]^{\frac{1}{2}}$
$\left[\frac{G h}{c}\right]^{\frac{1}{2}}$
$\left[\frac{G h}{c^{4}}\right]^{\frac{1}{2}}$
$\left[\frac{G h}{c^{3}}\right]^{\frac{1}{2}}$
Solution
To find the combination with dimensions of time, we check the dimensions for each:
1. Combination $\frac{G h}{c^5}$ gives $T^2$, not time.
2. Combination $\frac{G h}{c^2}$ gives $L^3 T^{-1}$, not time.
3. Combination $\frac{G h}{c^4}$ gives $L T$, not time.
4. Combination $\frac{G h}{c}$ gives $L^4 T^{-2}$, not time.
None of these combinations give time dimensions, so there seems to be a mistake in the options. The intended answer should match the correct dimensional analysis, but none provided are valid for time.