What is the dimensional formula for strain energy density?
\[
  {\text{A}}{\text{. }}\left[ {{{\text{M}}^1}{{\text{L}}^2}{{\text{T}}^{ - 3}}} \right] \\
  {\text{B}}{\text{. }}\left[ {{{\text{M}}^1}{{\text{L}}^2}{{\text{T}}^3}} \right] \\
  {\text{C}}{\text{. }}\left[ {{{\text{M}}^1}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right] \\
  {\text{D}}{\text{. }}\left[ {{{\text{M}}^1}{{\text{L}}^2}{{\text{T}}^{ - 2}}} \right] \\
 \]

Hint: Here, we will proceed by writing down the formula of strain energy. Then, by using this formula we will write the formula for strain energy density. Finally, we will apply dimensional analysis on both sides of the formula.

Step By Step Answer:
Formula Used- ${\text{U}} = \dfrac{1}{2}{\text{V}}\sigma \varepsilon $.

According to strain energy formula, we can write (provided the stress is directly proportional to the strain)

${\text{U}} = \dfrac{1}{2}{\text{V}}\sigma \varepsilon {\text{ }} \to {\text{(1)}}$

where U is the strain energy, V is the volume of the body, $\sigma $ denotes the stress and $\varepsilon $ is the strain

Strain energy density is defined as the strain energy per unit volume of the body

i.e., Strain energy density = $\dfrac{{{\text{Strain Energy}}}}{{{\text{Volume}}}} = \dfrac{{\text{U}}}{{\text{V}}}$

By taking volume of the body V from the RHS to the LHS of equation (1), we get

Strain energy density $\dfrac{{\text{U}}}{{\text{V}}} = \dfrac{1}{2}\sigma \varepsilon {\text{ }} \to {\text{(2)}}$

Stress is defined as the force applied or experienced per unit area

i.e., Stress $\sigma = \dfrac{{\text{F}}}{{\text{A}}}{\text{ }} \to {\text{(3)}}$ where F denotes the force applied or experienced and A denotes the area on which it is applied

As, dimensional formula for force is $\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]$ and that for area is $\left[ {{{\text{L}}^2}} \right]$

By applying dimensional analysis on both sides of equation (3), we get

Dimensional formula for stress =
\[\dfrac{{{\text{Dimensional formula for force}}}}{{{\text{Dimensional formula for area}}}} = \dfrac{{\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]}}{{\left[ {{{\text{L}}^2}} \right]}} = \left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\left[ {{{\text{L}}^{ - 2}}} \right] = \left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]\]

Also we know that strain is the ratio of the deformation produced after the application of a force on the body to the original dimension of the body

i.e., Strain = $\dfrac{{{\text{Deformation in dimension}}}}{{{\text{Original dimension}}}} = \dfrac{{\Delta {\text{L}}}}{{\text{L}}}$ where $\Delta {\text{L}}$denotes the change in length of the body and L denotes the original length of the body

Clearly, we can see from the formula of strain that it is a dimensionless quantity i.e., $\varepsilon $ is dimensionless

By applying dimensional analysis to equation (2), we get

Dimensional formula for stress energy density = (Dimensional formula for stress)(Dimensional formula for strain)

But since strain is dimensionless so we can write,

Dimensional formula for stress energy density = Dimensional formula for stress
$ \Rightarrow $ Dimensional formula for stress energy density = \[\left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right] = \left[ {{{\text{M}}^1}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]\]

Therefore, option C is correct.

Note: In this particular problem, when dimensional analysis is applied to the formula i.e., Stress energy density = $\dfrac{1}{2}\sigma \varepsilon $, $\dfrac{1}{2}$ is a number (constant) and the numbers are dimensionless so its dimension will automatically be neglected. That’s why the dimension of stress energy density is equal to the product of the dimensions of stress and strain.