Question

Using the method of dimension, show that 1 joule = 10⁷ erg.

Answer 1

Let us consider, 1 unit of SI unit of work gives n units of CGS units of work.

We know that, Dimensional formula for Work is

$[ML^{2}T^{-2}] \:$

Since irrespective of the unit, Work has same dimensions, We can write

$1[M _ {1}L _ {1}^{2}T _ {1} ^{-2}] \: \: = n[M _ {2}L _ {2}^{2}T _ {2} ^{-2}] \\ \\ n = \frac{[M _ {1}L _ {1}^{2}T _ {1} ^{-2}]}{[M _ {2}L _ {2}^{2}T _ {2} ^{-2}]}$

Here, 1 corresponds to SI, and 2 corresponds to CGS.

In SI, Units are kg, m, s while CGS has g, cm, s

$n = \frac{1kg}{g} \times ( \frac{m}{cm} ) ^{2} \times ( \frac{s}{s} ) ^{ - 2} \\ \\ n = \frac{ {10}^{3}g }{g} \times ( \frac{ {10}^{2} cm}{cm} ) ^{2} \times (1) \\ \\ n = {10}^{3} \times ( {10 ^{2} })^{2} \\ \\ n = {10}^{3} \times {10}^{4} \\ \\ n = 10 ^{7}$

From our earlier discussion,

1 SI unit of work gives n = 10^7 CGS units of work.

Hence, 1 Joule = 10^7 erg.