Question

The hubble constant has the dimensions of
A. time
$\text{B}\text{. tim}{{\text{e}}^{\text{-1}}}$
C. Length
D. Mass

Answer 1

Hint: Hubble constant is a proportionality constant. It is used in the equation to show the relation between the velocity of two galaxies (moving apart) and the distance between these two galaxies. It tells how our universe is expanding.

Complete step by step answer:
The size of the universe in which we live is not fixed. It is continuously expanding in all directions. It is expanding even while you are reading this page. Expansion of the universe does not mean that the contents of the universe are expanding as we usually see in our daily lives. Expanding universe means that the contents like planets, stars and galaxies, are continuously moving apart. That is the distance between galaxies is not fixed and the distance is always increasing.
For us it is as if the celestial bodies around us are moving away from earth. Now, if the galaxies are moving apart, they will have some velocity. Let that velocity be v and the distance between two galaxies be d. The relation between the velocity of two galaxies (v) and the distance between the galaxies be d is given by Hubble’s law i.e. $v={{H}_{o}}d$ …..(i).
${{H}_{o}}$ is a proportionality constant called Hubble constant. From equation (i), we understand that the velocity of galaxies is directly proportional to the distance between the galaxies. Farther the galaxies, faster the galaxies move apart.
From equation (i), we can find the dimension of the Hubble constant. We can write equation (i) as ${{H}_{o}}=\dfrac{d}{v}$. Let the dimension formula of ${{H}_{o}}$ be $\left[ {{H}_{o}} \right]$. We know that the dimensional formulas of velocity v and distance d are $\left[ v \right]=\left[ L{{T}^{-1}} \right]$ and $\left[ d \right]=\left[ L \right]$ respectively.
Therefore, $\left[ {{H}_{o}} \right]=\dfrac{\left[ d \right]}{\left[ v \right]}=\dfrac{\left[ L \right]}{\left[ L{{T}^{-1}} \right]}=\left[ {{T}^{-1}} \right]$.
Therefore, the dimension formula of Hubble constant is $\left[ {{T}^{-1}} \right]$.
Hence, the dimension of Hubble constant is $\text{tim}{{\text{e}}^{\text{-1}}}$.

So, the correct answer is “Option B”.

Note:
The Hubble constant is not actually constant. It changes with time. The reason we call it a constant is because the universe expands at the same rate at every location in the universe. But the expansion rate changes with time (i.e. the relation between v and d). Therefore, the value of Hubble constant changes with time.