Which of the following does not have the same dimension ?

Options

A. stress
B. Energy gradient
C. bulk modulus
D. energy density

Answer

To determine which of the given options does not have the same dimension, we will analyze the dimensions of each term mentioned: stress, energy gradient, bulk modulus, and energy density.

### Step-by-Step Solution:

1. **Calculate the Dimension of Stress:**
- Stress is defined as force per unit area.
- The dimension of force is given by:
\[ [\text{Force}] = [\text{mass}] \cdot [\text{acceleration}] = M^1 L^1 T^{-2} \]
- The dimension of area is:
\[ [\text{Area}] = L^2 \]
- Therefore, the dimension of stress is:
\[ [\text{Stress}] = \frac{[\text{Force}]}{[\text{Area}]} = \frac{M^1 L^1 T^{-2}}{L^2} = M^1 L^{-1} T^{-2} \]

2. **Calculate the Dimension of Energy Gradient:**
- Energy gradient is defined as energy per unit length.
- The dimension of energy is:
\[ [\text{Energy}] = M^1 L^2 T^{-2} \]
- Therefore, the dimension of energy gradient is:
\[ [\text{Energy Gradient}] = \frac{[\text{Energy}]}{[\text{Length}]} = \frac{M^1 L^2 T^{-2}}{L^1} = M^1 L^1 T^{-2} \]

3. **Calculate the Dimension of Bulk Modulus:**
- Bulk modulus is defined as bulk stress divided by bulk strain.
- Bulk stress has the same dimension as stress, which we calculated as \( M^1 L^{-1} T^{-2} \).
- Bulk strain is dimensionless.
- Therefore, the dimension of bulk modulus is:
\[ [\text{Bulk Modulus}] = \frac{[\text{Bulk Stress}]}{[\text{Bulk Strain}]} = M^1 L^{-1} T^{-2} \]

4. **Calculate the Dimension of Energy Density:**
- Energy density is defined as energy per unit volume.
- The dimension of volume is:
\[ [\text{Volume}] = L^3 \]
- Therefore, the dimension of energy density is:
\[ [\text{Energy Density}] = \frac{[\text{Energy}]}{[\text{Volume}]} = \frac{M^1 L^2 T^{-2}}{L^3} = M^1 L^{-1} T^{-2} \]

5. **Compare the Dimensions:**
- We have calculated the dimensions as follows:
- Stress: \( M^1 L^{-1} T^{-2} \)
- Energy Gradient: \( M^1 L^1 T^{-2} \)
- Bulk Modulus: \( M^1 L^{-1} T^{-2} \)
- Energy Density: \( M^1 L^{-1} T^{-2} \)

6. **Identify the Odd One Out:**
- The dimensions of stress, bulk modulus, and energy density are all \( M^1 L^{-1} T^{-2} \).
- The dimension of energy gradient is \( M^1 L^1 T^{-2} \).
- Therefore, the term that does not have the same dimension is **Energy Gradient**.

### Final Answer:
The option that does not have the same dimension is **Energy Gradient**.