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

Options

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

Explanation

To determine which of the given quantities has different dimensions from the others, we will analyze the dimensions of each option step by step.

### Given Quantities:
1. Energy Density
2. Force per Unit Area
3. Product of Charge per Unit Volume and Voltage
4. Angular Momentum per Unit Mass

### Step 1: Analyze Energy Density
- **Definition**: Energy density is defined as energy per unit volume.
- **Dimensions**:
- Energy has dimensions: \([M L^2 T^{-2}]\)
- Volume has dimensions: \([L^3]\)
- **Calculation**:
\[ \text{Energy Density} = \frac{\text{Energy}}{\text{Volume}} = \frac{[M L^2 T^{-2}]}{[L^3]} = [M L^{-1} T^{-2}] \]

### Step 2: Analyze Force per Unit Area
- **Definition**: Force per unit area is also known as pressure.
- **Dimensions**:
- Force has dimensions: \([M L T^{-2}]\)
- Area has dimensions: \([L^2]\)
- **Calculation**:
\[ \text{Force per Unit Area} = \frac{\text{Force}}{\text{Area}} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

### Step 3: Analyze Product of Charge per Unit Volume and Voltage
- **Definition**: This is the product of charge density and voltage.
- **Dimensions**:
- Charge per unit volume has dimensions: \([Q L^{-3}]\)
- Voltage (potential difference) is defined as work done per unit charge: \([M L^2 T^{-2}]/[Q]\)
- **Calculation**:
\[ \text{Voltage} = \frac{\text{Work}}{\text{Charge}} = \frac{[M L^2 T^{-2}]}{[Q]} \]
Therefore, the product of charge per unit volume and voltage is:
\[ \text{Product} = [Q L^{-3}] \cdot \left(\frac{[M L^2 T^{-2}]}{[Q]}\right) = [M L^{-1} T^{-2}] \]

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

### Conclusion
Now we can summarize the dimensions of all four quantities:
1. Energy Density: \([M L^{-1} T^{-2}]\)
2. Force per Unit Area: \([M L^{-1} T^{-2}]\)
3. Product of Charge per Unit Volume and Voltage: \([M L^{-1} T^{-2}]\)
4. Angular Momentum per Unit Mass: \([L^2 T^{-1}]\)

**The quantity with different dimensions is Angular Momentum per Unit Mass, which has dimensions \([L^2 T^{-1}]\).**