The dimension of `((1)/(2))epsilon_(0)E^(2)` (`epsilon_(0)` : permittivity of free space, E electric field

Options

A. (a) `MLT^(-1)`
B. (b) `ML^(2)T^(2)`
C. ( c ) `ML^(-1)T^(-2)`
D. ( d ) `ML^(2)T^(-1)`

Explanation

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

### Step 1: Understand the formula for energy density
The expression \(\frac{1}{2} \epsilon_0 E^2\) represents the energy density (energy per unit volume) stored in an electric field.

### Step 2: Identify the dimensions of energy
Energy can be defined as work done. The work done \(W\) is given by the formula:
\[ W = F \cdot d \]
where \(F\) is force and \(d\) is displacement. The dimension of force \(F\) is:
\[ [F] = M L T^{-2} \]
Thus, the dimension of work (or energy) becomes:
\[ [W] = [F] \cdot [d] = (M L T^{-2}) \cdot (L) = M L^2 T^{-2} \]

### Step 3: Identify the dimensions of volume
Volume \(V\) is given by:
\[ [V] = L^3 \]

### Step 4: Calculate the dimensions of energy density
Energy density is energy per unit volume:
\[ \text{Energy Density} = \frac{\text{Energy}}{\text{Volume}} = \frac{M L^2 T^{-2}}{L^3} = M L^{-1} T^{-2} \]

### Step 5: Relate energy density to \(\epsilon_0 E^2\)
From the definition of energy density in an electric field:
\[ \frac{1}{2} \epsilon_0 E^2 \]
we can conclude that the dimensions of \(\epsilon_0 E^2\) must also equal the dimensions of energy density, which we calculated as \(M L^{-1} T^{-2}\).

### Step 6: Conclusion
Thus, the dimension of \(\frac{1}{2} \epsilon_0 E^2\) is:
\[ [M L^{-1} T^{-2}] \]

### Final Answer
The dimension of \(\frac{1}{2} \epsilon_0 E^2\) is \(M L^{-1} T^{-2}\).

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