Of the following quantities , which one has the dimensions different from the remaining three?

Options

A. Energy per unit volume
B. Force per unit area
C. Product of voltage and charge per unit volume
D. Angular momentum per unit mass

Answer

To solve the problem of identifying which quantity has a different dimension from the others, we will analyze the dimensions of each option step by step.

### Step 1: Analyze Option A - Energy per Unit Volume
- **Definition**: Energy per unit volume is given by \( \frac{E}{V} \).
- **Dimensions**:
- Energy (E) has dimensions: \( [E] = M L^2 T^{-2} \)
- Volume (V) has dimensions: \( [V] = L^3 \)
- **Calculation**:
\[ \text{Dimensions of } \frac{E}{V} = \frac{M L^2 T^{-2}}{L^3} = M L^{-1} T^{-2} \]

### Step 2: Analyze Option B - Force per Unit Area
- **Definition**: Force per unit area is given by \( \frac{F}{A} \).
- **Dimensions**:
- Force (F) has dimensions: \( [F] = M L T^{-2} \)
- Area (A) has dimensions: \( [A] = L^2 \)
- **Calculation**:
\[ \text{Dimensions of } \frac{F}{A} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]

### Step 3: Analyze Option C - Product of Voltage and Charge per Unit Volume
- **Definition**: Voltage (V) multiplied by charge (Q) per unit volume.
- **Dimensions**:
- Voltage (V) has dimensions: \( [V] = M L^2 T^{-3} I^{-1} \)
- Charge (Q) has dimensions: \( [Q] = I T \)
- Volume (V) has dimensions: \( [V] = L^3 \)
- **Calculation**:
\[ \text{Dimensions of } V \cdot Q = (M L^2 T^{-3} I^{-1}) \cdot (I T) = M L^2 T^{-2} \]
\[ \text{Dimensions of } \frac{V \cdot Q}{V} = \frac{M L^2 T^{-2}}{L^3} = M L^{-1} T^{-2} \]

### Step 4: Analyze Option D - Angular Momentum per Unit Mass
- **Definition**: Angular momentum (L) per unit mass (M).
- **Dimensions**:
- Angular momentum has dimensions: \( [L] = M L^2 T^{-1} \)
- Mass (M) has dimensions: \( [M] = M \)
- **Calculation**:
\[ \text{Dimensions of } \frac{L}{M} = \frac{M L^2 T^{-1}}{M} = L^2 T^{-1} \]

### Conclusion
Now, we can summarize the dimensions of all options:
- Option A: \( M L^{-1} T^{-2} \)
- Option B: \( M L^{-1} T^{-2} \)
- Option C: \( M L^{-1} T^{-2} \)
- Option D: \( L^2 T^{-1} \)

Since the dimensions of options A, B, and C are the same, while option D has a different dimension, the answer is:

**Option D has a different dimension from the other three.**