]Which of the following is the dimensions of `1/2 epsilon_0 E^2`

Options

A. `ML^(-3) T^(-1)`
B. `ML^(1)T^(-2)`
C. `ML^(-1)T^(-2)`
D. `ML^2T^(-2)`

Explanation

To find the dimensions of the expression \( \frac{1}{2} \epsilon_0 E^2 \), where \( \epsilon_0 \) is the permittivity of free space and \( E \) is the electric field, we can follow these steps:

### Step 1: Understand the Expression
The expression \( \frac{1}{2} \epsilon_0 E^2 \) represents the energy density of an electric field in a capacitor. Energy density is defined as energy per unit volume.

### Step 2: Identify Dimensions of Energy
The dimensions of energy can be derived from the formula for work done, which is given by:
\[ \text{Work} = \text{Force} \times \text{Displacement} \]
The dimensional formula for force is:
\[ \text{Force} = \text{mass} \times \text{acceleration} = m \cdot L \cdot T^{-2} \]
Thus, the dimensions of energy (or work) are:
\[ \text{Energy} = \text{Force} \times \text{Displacement} = (m \cdot L \cdot T^{-2}) \cdot L = m \cdot L^2 \cdot T^{-2} \]

### Step 3: Identify Dimensions of Volume
The dimensions of volume are given by:
\[ \text{Volume} = \text{length}^3 = L^3 \]

### Step 4: Calculate Dimensions of Energy Density
Energy density is defined as energy per unit volume:
\[ \text{Energy Density} = \frac{\text{Energy}}{\text{Volume}} = \frac{m \cdot L^2 \cdot T^{-2}}{L^3} = m \cdot L^{-1} \cdot T^{-2} \]

### Step 5: Conclusion
Thus, the dimensions of \( \frac{1}{2} \epsilon_0 E^2 \) are:
\[ [m \cdot L^{-1} \cdot T^{-2}] \]

### Final Answer
The dimensions of \( \frac{1}{2} \epsilon_0 E^2 \) are \( [M^1 L^{-1} T^{-2}] \).

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