Question

The term Planck time \[{{t}_{P}}\] refers to the time after the ‘Big Bang’, at which the laws of physics began to be applied to understand the physical phenomenon. Check whether the proposed formula for Planck time, given below, is correct or not?
\[{{t}_{P}}=\dfrac{1}{\sqrt{2\pi }}\sqrt{\dfrac{GE}{\nu {{c}^{5}}}}\]
Here G, E, c and ν corresponds to the universal gravitational constant, energy, the speed of light and frequency respectively.

Answer 1

Hint: We have given a formula for Planck time and we have to verify it. It is also given that Planck time refers to time so it will have the same dimension of time. Therefore by doing dimensional analysis of the given formula we can verify whether the equation is correct or not. We will directly substitute the dimension of G, E, c and ν in the given formula.

Complete answer:
Planck time is the time period taken by a photon to cover a distance of Planck length. We have given the formula of Planck time in terms of universal gravitational constant G, energy E, speed of light c and frequency ν.
\[{{t}_{P}}=\dfrac{1}{\sqrt{2\pi }}\sqrt{\dfrac{GE}{\nu {{c}^{5}}}}\]
By using dimensions of G, E, c and ν we can verify the formula. As according to the rule of dimensional analysis both sides of the equation should have the same dimension. The Planck time will have dimension of time as the question says that Planck time refers to time.
Dimension of universal gravitational constant is \[\left[ {{M}^{-1}}{{L}^{3}}{{T}^{-2}} \right]\], dimension of energy is \[\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right]\], speed of light has dimension \[\left[ {{L}^{1}}{{T}^{-1}} \right]\]and frequency has dimension \[\left[ {{T}^{-1}} \right]\].
Now substituting this dimensions in the given formula in place of its respected values we have,
\[\begin{align}
  & \left[ T \right]=\sqrt{\dfrac{\left[ {{M}^{-1}}{{L}^{3}}{{T}^{-2}} \right]\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}} \right]}{\left[ {{T}^{-1}} \right]{{\left[ {{L}^{1}}{{T}^{-1}} \right]}^{5}}}}\text{ }\left( \because \dfrac{1}{\sqrt{2\pi }}\text{ is dimensionless quantity} \right) \\
 & \Rightarrow \left[ T \right]=\sqrt{\dfrac{\left[ {{M}^{0}}{{L}^{5}}{{T}^{-4}} \right]}{\left[ {{T}^{-1}} \right]\left[ {{L}^{5}}{{T}^{-5}} \right]}} \\
 & \Rightarrow \left[ T \right]=\sqrt{\dfrac{\left[ {{M}^{0}}{{L}^{5}}{{T}^{-4}} \right]}{\left[ {{L}^{5}}{{T}^{-6}} \right]}} \\
 & \Rightarrow \left[ T \right]=\sqrt{\left[ {{M}^{0}}{{L}^{5}}{{T}^{-4}} \right]\left[ {{L}^{-5}}{{T}^{6}} \right]} \\
 & \Rightarrow \left[ T \right]=\sqrt{\left[ {{M}^{0}}{{L}^{0}}{{T}^{2}} \right]} \\
 & \Rightarrow \left[ T \right]=T \\
\end{align}\]
As we can see that both sides of the formula have the same dimension, hence we conclude that the formula is correct.

Note:
We can also verify the dimension of universal gravitational constant G and energy E the same way by using gravitational force formula and mass-energy relationship respectively. We can derive the dimensions of the physical quantity by its units and vice-versa. π is measured in radians which is dimensionless therefore \[\dfrac{1}{\sqrt{2\pi }}\] is also dimensionless quantity