Understanding Dimensions
To determine the dimension that matches a certain physical quantity, it is essential to analyze each option provided: energy per unit volume, energy, force per unit volume, and force.
1. Energy per unit volume
- Energy (E) has the dimension of [M^1 L^2 T^-2].
- Volume (V) has the dimension of [L^3].
- Therefore, energy per unit volume can be expressed as:
\[ \text{Energy per unit volume} = \frac{E}{V} = \frac{[M^1 L^2 T^{-2}]}{[L^3]} = [M^1 L^{-1} T^{-2}] \]
2. Energy
- The dimension of energy remains as:
\[ [M^1 L^2 T^{-2}] \]
3. Force per unit volume
- Force (F) has the dimension of [M^1 L^1 T^-2].
- Thus, force per unit volume is given by:
\[ \text{Force per unit volume} = \frac{F}{V} = \frac{[M^1 L^1 T^{-2}]}{[L^3]} = [M^1 L^{-2} T^{-2}] \]
4. Force
- The dimension of force is:
\[ [M^1 L^1 T^{-2}] \]
Conclusion
Comparing the dimensions calculated:
- Energy per unit volume: [M^1 L^{-1} T^{-2}]
- Energy: [M^1 L^2 T^{-2}]
- Force per unit volume: [M^1 L^{-2} T^{-2}]
- Force: [M^1 L^1 T^{-2}]
The only dimension that matches with energy per unit volume is Force per unit volume, which has the same dimensional structure of [M^1 L^{-2} T^{-2}].
Thus, the dimension that is the same as energy per unit volume is force per unit volume.