The energy density and pressure have

Options

A. Same dimensions
B. Different dimensions
C. No dimensions
D. None

Answer

To solve the question regarding the relationship between energy density and pressure, we will derive the dimensional formulas for both quantities and compare them.

### Step-by-Step Solution:

**Step 1: Understand Energy Density**
- Energy density (u) is defined as the amount of energy (E) per unit volume (V).
- The formula for energy density is:
\[ u = \frac{E}{V} \]

**Step 2: Find the Dimensional Formula for Energy**
- The dimensional formula for energy (E) is:
\[ [E] = [M^1 L^2 T^{-2}] \]
where M is mass, L is length, and T is time.

**Step 3: Find the Dimensional Formula for Volume**
- Volume (V) is given by the formula for a cube, which is length cubed:
\[ [V] = [L^3] \]

**Step 4: Calculate the Dimensional Formula for Energy Density**
- Substitute the dimensional formulas for energy and volume into the energy density formula:
\[ [u] = \frac{[E]}{[V]} = \frac{[M^1 L^2 T^{-2}]}{[L^3]} = [M^1 L^{-1} T^{-2}] \]

**Step 5: Understand Pressure**
- Pressure (P) is defined as force (F) per unit area (A).
- The formula for pressure is:
\[ P = \frac{F}{A} \]

**Step 6: Find the Dimensional Formula for Force**
- The dimensional formula for force (F) is derived from Newton's second law (F = ma):
\[ [F] = [M^1 L^1 T^{-2}] \]

**Step 7: Find the Dimensional Formula for Area**
- Area (A) is given by length squared:
\[ [A] = [L^2] \]

**Step 8: Calculate the Dimensional Formula for Pressure**
- Substitute the dimensional formulas for force and area into the pressure formula:
\[ [P] = \frac{[F]}{[A]} = \frac{[M^1 L^1 T^{-2}]}{[L^2]} = [M^1 L^{-1} T^{-2}] \]

**Step 9: Compare the Dimensional Formulas**
- From the calculations:
- Dimensional formula of energy density: \([M^1 L^{-1} T^{-2}]\)
- Dimensional formula of pressure: \([M^1 L^{-1} T^{-2}]\)

Since both energy density and pressure have the same dimensional formula, we conclude that they have the same dimensions.

### Final Answer:
The energy density and pressure have the **same dimension**.

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