Question

The smallest meaningful measure of length is called the Planck length, and it is defined in terms of three fundamental constants in nature: the speed of light , the gravitational constant $\mathbf{G}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{67}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}} {\mathbf{m}}^{\mathbf{3}}\mathbf{\times }{\mathbf{kg}}^{\mathbf{-}\mathbf{1}}\mathbf{\times }{\mathbf{s}}^{\mathbf{-}\mathbf{2}}$, and Planck’s constant $\mathbf{h}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{63}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{34}} {\mathbf{m}}^{\mathbf{2}}\mathbf{\times }\mathbf{kg}\mathbf{\times }{\mathbf{s}}^{\mathbf{-}\mathbf{1}}$. The Planck length ${\ell }_{\mathrm{p}}$is given by the following combination of these three constants:

${\ell }_{\mathrm{p}}=\sqrt{\frac{\mathrm{Gh}}{{\mathrm{c}}^3}}$

Show that the dimensions of${\ell }_{\mathrm{p}}$are length [L], and find the order of magnitude of${\ell }_{\mathrm{p}}$. [Recent theories (Chapters 32 and 33) suggest that the smallest particles (quarks, leptons) are ‘strings’ with lengths on the order of the Planck length,${\mathbf{10}}^{\mathbf{-}\mathbf{35}} \mathbf{m}$. These theories also suggest that the ‘Big Bang’, with which the universe is believed to have begun, started from an initial size on the order of the Planck length.]

Short answer

The dimension of Planck’s is [L], and that of the order of magnitude is ${10}^{-35} \mathrm{m}$.

Step-by-step solution

Step 1. Identification of the given data

Given data:

The speed of light,$\mathrm{c}=3.00\times {10}^8 \mathrm{m}·{\mathrm{s}}^{-1}.$

The gravitational constant,$\mathrm{G}=6.67\times {10}^{-11} {\mathrm{m}}^3·{\mathrm{kg}}^{-1}·{\mathrm{s}}^{-2}.$

The Planck’s constant,$\mathrm{h}=6.63\times {10}^{-34} {\mathrm{m}}^2·\mathrm{kg}·{\mathrm{s}}^{-1}.$

The equation of Planck’s length,${\ell }_{\mathrm{p}}=\sqrt{\frac{\mathrm{Gh}}{{\mathrm{c}}^3}}.$

Step 2. Determining the dimensional formula for the different quantities

You know that the unit of parameters on both sides of an equation will be the same. In other words, you can equate parameters having the same dimensions (or units).

From the given units of the speed of light, the gravitational constant, and Planck’s constant, you can find out the dimensions of each parameter. The unit of the speed of light is $\mathrm{m}·{\mathrm{s}}^{-1}.$Thus, its dimension will be$\left[\mathrm{c}\right]=\left[{\mathrm{M}}^0 {\mathrm{L}}^1 {\mathrm{T}}^{-1}\right].$

In the case of the gravitational constant, the unit is ${\mathrm{m}}^3·{\mathrm{kg}}^{-1}·{\mathrm{s}}^{-2}$. Thus, the dimension is

$\left[\mathrm{G}\right]=\left[{\mathrm{M}}^{-1} {\mathrm{L}}^3 {\mathrm{T}}^{-2}\right]$.

Similarly, the unit of the Planck’s constant is ${\mathrm{m}}^2·\mathrm{kg}·{\mathrm{s}}^{-1}$. Then, the dimension is

$\left[\mathrm{h}\right]=\left[{\mathrm{M}}^1 {\mathrm{L}}^2 {\mathrm{T}}^{-1}\right]$.

Calculating the order of magnitude.

Applying all the dimensions in the equation of Planck’s length to find the dimension of Planck’s length,

$\left[{\ell }_{\mathrm{p}}\right]=\sqrt{\frac{\left[\mathrm{G}\right]\left[\mathrm{h}\right]}{{\left[\mathrm{c}\right]}^3}}$.

Applying the values,

$\begin{array}{c}\left[{\ell }_{\mathrm{p}}\right]={\left(\frac{\left[{\mathrm{M}}^{-1} {\mathrm{L}}^3 {\mathrm{T}}^{-2}\right]\times \left[{\mathrm{M}}^1 {\mathrm{L}}^2 {\mathrm{T}}^{-1}\right]}{{\left[{\mathrm{M}}^0 {\mathrm{L}}^1 {\mathrm{T}}^{-1}\right]}^3}\right)}^{\frac{1}{2}}\\ ={\left(\left[{\mathrm{M}}^{\left(-1+1\right)} {\mathrm{L}}^{\left(5-3\right)} {\mathrm{T}}^{\left(-3+3\right)}\right]\right)}^{\frac{1}{2}}\\ ={\left[{\mathrm{M}}^0 {\mathrm{L}}^2 {\mathrm{T}}^0\right]}^{\frac{1}{2}}\\ =\left[{\mathrm{M}}^0 {\mathrm{L}}^1 {\mathrm{T}}^0\right]\end{array}$.

You can find out the order of the magnitude of Planck’s length by applying all the values in the equation. So,

$\begin{array}{c}{\ell }_{\mathrm{p}}=\sqrt{\frac{6.67\times {10}^{-11} \times 6.63\times {10}^{-34} }{{\left(3.00\times {10}^8\right)}^3}}\\ =1.616\times {10}^{-35 } \mathrm{m}\end{array}$

So, the order of magnitude is ${10}^{-35} \mathrm{m}$.