Question

Convert $1Joule$ to $ergs$ in dimensional analysis.

Answer 1

First of all, we have to understand the meaning and use of unit joules according to the guidelines of the SI system of units. Joules is the unit of work or energy and this is equal to the Force applied in unit newton through unit meter. Secondly, for dimensional analysis we have to use their dimensions and convert joules to ergs.
Dimensions of joules as well as ergs, since both are the units of energy: $[{M^1}{L^2}{T^{ - 2}}]$.

Complete step by step solution:
Here, we have to convert joules to ergs as joules is the SI unit of energy and ergs is the CGS unit of energy.
Now, let us use the dimensional formula for joules and ergs as follows:
${n_1}joules = {n_2}ergs$ …. $(1)$
Let us use dimensions of joules and ergs as:
$joule = [M_1^1L_1^2T_1^{ - 2}]$ and $erg = [M_2^1L_2^2T_2^{ - 2}]$
Using these values in equation $(1)$ such that:
${n_1}[M_1^1L_1^2T_1^{ - 2}] = {n_2}[M_2^1L_2^2T_2^{ - 2}]$
$\Rightarrow {n_2} = {n_1}\frac{{[M_1^1L_1^2T_1^{ - 2}]}}{{[M_2^1L_2^2T_2^{ - 2}]}}$ …. $(2)$
$\Rightarrow {n_2} = {n_1}{\left[ {\frac{{{M_1}}}{{{M_2}}}} \right]^1}{\left[ {\frac{{{L_1}}}{{{L_2}}}} \right]^2}{\left[ {\frac{{{T_1}}}{{{T_2}}}} \right]^{ - 2}}$
$\Rightarrow {n_2} = 1{\left[ {\frac{{1kg}}{{gm}}} \right]^1}{\left[ {\frac{{1m}}{{1cm}}} \right]^2}{\left[ {\frac{{1s}}{{1s}}} \right]^{ - 2}}$
$\Rightarrow {n_2} = 1{\left[ {\frac{{1000gm}}{{gm}}} \right]^1}{\left[ {\frac{{100cm}}{{1cm}}} \right]^2}{\left[ {\frac{{1s}}{{1s}}} \right]^{ - 2}}$
$\Rightarrow {n_2} = {10^7}(ergs)$ …. ${\text{(3)}}$
Now, we know that
$1{\text{ joule of energy}} = 1{\text{ newton}} \times 1{\text{ meter}}$
${\text{1 erg = 1 dyne}} \times {\text{1 cm}}$
${\text{1 joule = 1}}{{\text{0}}^5}{\text{ dynes }} \times {\text{ 1}}{{\text{0}}^2}{\text{ cm}}$
${\text{1 joule = 1}}{{\text{0}}^7}{\text{ ergs}}$ …. From ${\text{(3)}}$
Thus, this is the way to convert $1Joule$ to $ergs$ with the help of dimensional analysis. We conclude that ${\text{1 joule = 1}}{{\text{0}}^7}{\text{ ergs}}$.

Note:
Here, we must know the dimensional formulae of joules and ergs are the same because they are the SI and CGS units respectively of energy and work done. So, it will be easier to convert them from SI to CGS and find the required answer. Also, we have to convert mass and lengths from SI to CGS so that the whole joule can be converted to ergs as we have discussed above.