Question

If the velocity of light (c), acceleration due to gravity (g), and atmospheric pressure (p) are chosen as fundamental units, then the dimensions of length will be?

Answer 1

Step 1: Given parameters

1. The velocity of light = $c$
2. Acceleration due to gravity = $g$
3. Atmospheric pressure = $p$

Step 2: Formula used

Use the following expression to find the dimensions for length.

$L=c^x{g}^y{p}^z...\left(1\right)$

Where, $x$,$y$, and $z$ are the powers to which $c$, $g$, and $p$ are raised.

The dimensions for the given quantities are as follows:

1. Dimension of the velocity of light, $c$ = $\left[M^0{L}^1{T}^{-1}\right]...\left(2\right)$
2. Dimension of acceleration due to gravity, $g$ = $\left[M^0{L}^1{T}^{-2}\right]...\left(3\right)$
3. Dimension of atmospheric pressure, $p$ = $\left[M^1{L}^{-1}{T}^{-2}\right]...\left(4\right)$

Step 3: Calculating dimension

Substitute equations (2), (3), and (4) in equation (1).

$$\begin{gathered} \left[M^0{L}^1{T}^0\right]={\left[M^0{L}^1{T}^{-1}\right]}^x{\left[M^0{L}^1{T}^{-2}\right]}^y{\left[M^1{L}^{-1}{T}^{-2}\right]}^z \\ \left[M^0{L}^1{T}^0\right]=\left[M^z{L}^{x+y-z}{T}^{-x-2y-2z}\right] \end{gathered}$$

Apply the principle of homogeneity,

Comparing the powers on both sides, we get :

Comparing the powers of $M$ and $L$,

$$\begin{gathered} z=0 \\ x+y-z=1 \\ Asz=0 \\ \Rightarrow x+y=1 \\ \Rightarrow x=1-y...\left(5\right) \end{gathered}$$

Comparing the powers of $T$,

$$\begin{gathered} -x-2y-2z=0 \\ Asz=0,so \\ -x-2y=0...\left(6\right) \end{gathered}$$

Substitute equation (5) in equation (6).

$$\begin{gathered} -(1-y)-2y=0 \\ -1+y-2y=0 \\ -1-y=0 \\ \Rightarrow y=-1 \end{gathered}$$

Substitute value of $y$ in equation (5).

$$\begin{gathered} x=1-y \\ =1-\left(-1\right) \\ x=2 \end{gathered}$$

Substitute the values of $x$,$y$, and $z$ in equation (1).

$$\begin{gathered} L=c^x{g}^y{p}^z \\ L=c^2{g}^{-1}{p}^0 \\ L=\frac{c^2}{g} \end{gathered}$$

Hence, if the velocity of light, acceleration due to gravity, and atmospheric pressure are chosen as fundamental units, then the dimensions of length will be $L=\frac{c^2}{g}$.