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# The ratio of SI units to CGS units of energy isA. ${10^5}$B. ${10^6}$C. ${10^7}$D. ${10^{ - 7}}$

Last updated date: 06th Sep 2024
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Hint: Concept conversion of one system of units into another as SI and CGS are the two different systems of units. To find the ratio, one must know their inter conversion.
${n_1}{u_1} = {n_2}{u_2} \\ \Rightarrow {n_1}\left[ {M_1^a{\text{ }}L_1^b{\text{ }}T_1^C} \right] = {n_2}\left[ {M_2^a{\text{ }}L_2^b{\text{ }}T_2^c} \right] \\$

Complete step by step answer:
$\to$ Conversion of one system of units into another is based upon the fact that the magnitude of a physical quantity remains the same, whatever may be the system of units.
$\to$ We know that SI units of energy are joule (J) and cgs units of energy are erg
$\to$ Dimensional formula of energy is
Energy $= \dfrac{1}{2}m{v^2}$
$= \left[ M \right]{\left[ {L{T^{ - 1}}} \right]^2}$…… (as$V = \dfrac{d}{t}$$v = \dfrac{L}{T} = L{T^{ - 1}}$)
Energy $= \left[ {M{L^2}{T^{ - 2}}} \right]$
Now, ${n_1}\left[ {M_1^a{\text{ L}}_1^b{\text{ T}}_1^c} \right] = {n_2}\left[ {M_2^a{\text{ L}}_2^b{\text{ T}}_2^c} \right]$ …. (i)
Here, $a = 1,{\text{ b}} = 2,{\text{ c}} = - 2$
Here, $\left[ M \right]$ represent the dimensional formula of mass
$\left[ L \right]$ Represents the dimensional formula of length $\left[ T \right]$ represent the dimensional formula of time
 SI CGS ${M_1} = 1kg = 1000g$ ${M_2} = 1g$ ${L_1} = 1m = 100cm$ ${L_2} = 1cm$ ${T_1} = 1\sec$ ${T_2} = 1\sec$ ${n_1} = 1$(joule) ${n_2} = ?$ erg

Putting all these values in equation (i), we get
${n_1}\left[ {M_1^a{\text{ L}}_1^b{\text{ T}}_1^c} \right] = {n_2}\left[ {M_2^a{\text{ L}}_2^b{\text{ T}}_2^c} \right] \\ {n_2} = {n_1}\left[ {\dfrac{{{M_1}}}{{{M_2}}}} \right]_2^a\left[ {\dfrac{{{L_1}}}{{{L_2}}}} \right]_{}^b{\left[ {\dfrac{{{T_1}}}{{{T_2}}}} \right]^c} \\ {n_2} = 1{\left[ {\dfrac{{1000g}}{{1g}}} \right]^1}{\left[ {\dfrac{{100cm}}{{1cm}}} \right]^2}{\left[ {\dfrac{{1\sec }}{{1\sec }}} \right]^{ - 2}} \\ {n_2} = 1 \times 1000 \times {\left( {100} \right)^2} \times {\left( 1 \right)^{ - 2}} \\ {n_2} = {10^7} \\$
On solving this, we have
So, $1$ Joule $= {10^7}$ ergs
Ration of SI to CGS $= \dfrac{{Joule}}{{erg}}$
Substituting the values of joules in ergs
$= \dfrac{{{{10}^7}erg}}{{erg}} \\ = {10^7} \\$

Note:
Remember that they have asked for a ratio of SI units of energy to the CGS units of energy, so the correct option is ${10^7}$ not${10^{ - 7}}$. Also, the dimensional formula of work done and all energies are the same.