Question

Find the dimensional formula for hc/G?

Answer 1

Here, the given terms are:

$H$ is plank's constant, $c$ is the speed of light and $G$ is gravitational constant.

we know, the energy of a photon is given as:

$E=\frac{hc}{wavelength}$

the dimension of hc = dimension of energy $\times$ length

dimension of energy =$\left[M{L}^2{T}^{-2}\right]$

dimension of wavelength$=\left[L\right]$

so, dimension of $hc=\left[M{L}^2{T}^{-2}\right]\left[L\right]=\left[M{L}^3{T}^{-2}\right]$

now, dimension of G :

we know, gravitational force ,

$F=\frac{GMm}{r^2}$

so, dimension of $G=F\times \frac{r^2}{Mm}$

dimension of force$=\left[ML{T}^{-2}\right]$

dimension of $r^2=\left[L^2\right]$

dimension of $M=\left[M\right]$

dimension of $m=\left[M\right]$

so, dimension of G$=\left[ML{T}^{-2}\right]\left[L^2\right]/\left[M\right]\left[M\right]$

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

now, dimension of hc/G = dimension of hc/dimension of G

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

$=\left[M^2\right]$