Question

Using dimension show that $1$ joule $={10}^{7}erg$

Answer 1

It is given that $1joule={10}^{7}erg$

Both erg and joule are the units of energy in CGS and MKS respectively. The dimensional formula of energy is $\left[M{L}^2{T}^{-2}\right]$

Let

$1joule=Xerg$

$1\left[M_1{L_1}^2{T_1}^{-2}\right]=X\left[M_2{L_2}^2{T_2}^{-2}\right]$

$X=\frac{\left[M_1{L_1}^2{T_1}^{-2}\right]}{\left[M_2{L_2}^2{T_2}^{-2}\right]}$

$X=\left[\frac{M_1}{M_2}\right]{\left[\frac{L_1}{L_2}\right]}^2{\left[\frac{T_1}{T_2}\right]}^2$

$X=\left[\frac{Kg}{gm}\right]{\left[\frac{m}{cm}\right]}^2{\left[\frac{s}{s}\right]}^{-2}$

$X=\left[\frac{1000gm}{gm}\right]{\left[\frac{100cm}{cm}\right]}^2{\left(1\right)}^{-2}$

$={\left(10\right)}^3\left({10}^4\right)$

$={\left(10\right)}^7$

So, $1Joule={10}^{7}erg$