Question

The speed of light (c), gravitational constant (G) and Planck’s constant (h) are taken as fundamental units in a system. The dimensions of time in this new system should be:
$\begin{aligned} & (A){{G}^{\dfrac{1}{2}}}{{h}^{\dfrac{1}{2}}}{{c}^{-\dfrac{5}{2}}} \\\ & (B){{G}^{-\dfrac{1}{2}}}{{h}^{\dfrac{1}{2}}}{{c}^{\dfrac{1}{2}}} \\\ & (C){{G}^{\dfrac{1}{2}}}{{h}^{\dfrac{1}{2}}}{{c}^{-\dfrac{3}{2}}} \\\ & (D){{G}^{\dfrac{1}{2}}}{{h}^{\dfrac{1}{2}}}{{c}^{\dfrac{1}{2}}} \\\ \end{aligned}$

Answer 1

We will first relate the new dimensions of time as a linear relation between the speed of light, gravitational constant and Planck’s constant. Once, we do this, we can use the original dimensions of these terms in our equation and compare it with the original dimensions of time to get the required solution.

Complete step-by-step answer:
Let us first see the dimensional formula of these three terms.
Speed of light (c): $[{{M}^{0}}{{L}^{1}}{{T}^{-1}}]$
The gravitational constant (G): $[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]$
And lastly, the Planck’s constant (h): $[{{M}^{1}}{{L}^{2}}{{T}^{-1}}]$
Now, we can relate the dimensions of time in terms of these parameters as follows:
$\Rightarrow \left[ {{M}^{0}}{{L}^{0}}{{T}^{1}} \right]={{\left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]}^{x}}{{\left[ {{M}^{-1}}{{L}^{3}}{{T}^{-2}} \right]}^{y}}{{\left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right]}^{z}}$
Where, ‘x’, ‘y’ and ‘z’ are some arbitrary constants.
On simplify the above equation, we get:
$\Rightarrow \left[ {{M}^{0}}{{L}^{0}}{{T}^{1}} \right]=\left[ {{M}^{-y+z}}{{L}^{x+3y+2z}}{{T}^{-x-2y-z}} \right]$
Thus, we get three equations with three variables:
$\Rightarrow -y+z=0$ [Let this expression be equation number (1)]
$\Rightarrow x+3y+2z=0$ [Let this expression be equation number (2)]
$\Rightarrow -x-2y-z=1$ [Let this expression be equation number (3)]
Putting, $y=z$, from equation (1) into equation (2) and (3) and adding the resultant equations, we get:
$\begin{aligned} & \Rightarrow x+5y-x-3y=1 \\\ & \Rightarrow 2y=1 \\\ & \therefore y=\dfrac{1}{2} \\\ \end{aligned}$
Therefore, from equation (1), we get:
$\Rightarrow z=\dfrac{1}{2}$
Putting the value of ‘y’ and ‘z’ in equation number (2). We get the ‘x’ as:
$\begin{aligned} & \Rightarrow x+\dfrac{3}{2}+\dfrac{2}{2}=0 \\\ & \therefore x=-\dfrac{5}{2} \\\ \end{aligned}$
Thus, we can write:
$\Rightarrow \left[ Time \right]=\left[ {{c}^{-\dfrac{5}{2}}}{{G}^{\dfrac{1}{2}}}{{h}^{\dfrac{1}{2}}} \right]$
Hence, the new dimension of time (T) in terms of speed of light (c), Gravitational constant (G) and Planck’s constant (h) comes out to be $\left[ {{c}^{-\dfrac{5}{2}}}{{G}^{\dfrac{1}{2}}}{{h}^{\dfrac{1}{2}}} \right]$.

So, the correct answer is “Option A”.

Note: In problems like these, we should always find the dimensions of new fundamental units in terms of previous fundamental units as every other parameter is to be written in their terms. Also while solving these linear equations in higher variable terms, one should be careful at each step of calculation, so that we do not concur any mistakes in our solution.