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

Options

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

Explanation

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

### Step 1: Analyze Quantity A - Energy per Unit Volume
- **Energy (E)** has the dimension:
\[ [E] = [M^1 L^2 T^{-2}] \]
- **Volume (V)** has the dimension:
\[ [V] = [L^3] \]
- Therefore, the dimension of energy per unit volume is:
\[ \frac{[E]}{[V]} = \frac{[M^1 L^2 T^{-2}]}{[L^3]} = [M^1 L^{-1} T^{-2}] \]

### Step 2: Analyze Quantity B - Force per Unit Area
- **Force (F)** has the dimension:
\[ [F] = [M^1 L^1 T^{-2}] \]
- **Area (A)** has the dimension:
\[ [A] = [L^2] \]
- Therefore, the dimension of force per unit area is:
\[ \frac{[F]}{[A]} = \frac{[M^1 L^1 T^{-2}]}{[L^2]} = [M^1 L^{-1} T^{-2}] \]

### Step 3: Analyze Quantity C - Product of Charge per Unit Volume and Voltage
- **Charge (Q)** has the dimension:
\[ [Q] = [Q] \]
- **Voltage (V)** is defined as work per unit charge:
\[ [V] = \frac{[E]}{[Q]} = \frac{[M^1 L^2 T^{-2}]}{[Q]} \]
- The dimension of charge per unit volume is:
\[ \frac{[Q]}{[L^3]} \]
- Therefore, the dimension of the product of charge per unit volume and voltage is:
\[ \frac{[Q]}{[L^3]} \cdot \frac{[M^1 L^2 T^{-2}]}{[Q]} = \frac{[M^1 L^2 T^{-2}]}{[L^3]} = [M^1 L^{-1} T^{-2}] \]

### Step 4: Analyze Quantity D - Angular Momentum
- **Angular Momentum (L)** is defined as:
\[ L = R \cdot P \]
where \( R \) is the radius (dimension \( [L] \)) and \( P \) is momentum (dimension \( [M^1 L^1 T^{-1}] \)).
- Therefore, the dimension of angular momentum is:
\[ [L] = [L] \cdot [M^1 L^1 T^{-1}] = [M^1 L^2 T^{-1}] \]

### Conclusion
- The dimensions of the quantities are as follows:
- Quantity A: \([M^1 L^{-1} T^{-2}]\)
- Quantity B: \([M^1 L^{-1} T^{-2}]\)
- Quantity C: \([M^1 L^{-1} T^{-2}]\)
- Quantity D: \([M^1 L^2 T^{-1}]\)

Thus, the quantity with different dimensions is **Quantity D (Angular Momentum)**.