Find the dimensions of energy density.

Explanation

To find the dimensions of energy density, we can follow these steps:

### Step 1: Understand the definition of energy density
Energy density is defined as energy per unit volume. Mathematically, it can be expressed as:
\[ \text{Energy Density} = \frac{\text{Energy}}{\text{Volume}} \]

### Step 2: Identify the dimensions of energy
Energy can be defined in terms of work done. Work done (W) is given by the formula:
\[ W = \text{Force} \times \text{Displacement} \]
The dimensions of force (F) can be expressed as:
\[ F = \text{mass} \times \text{acceleration} = m \cdot a \]
Where:
- Mass (m) has dimensions [M]
- Acceleration (a) has dimensions [L T^{-2}]

Thus, the dimensions of force are:
\[ [F] = [M][L][T^{-2}] \]

Now substituting this into the work done:
\[ W = F \cdot d = (m \cdot a) \cdot d = (m \cdot L T^{-2}) \cdot L = m \cdot L^2 T^{-2} \]

So, the dimensions of energy (or work done) are:
\[ [W] = [M][L^2][T^{-2}] \]

### Step 3: Identify the dimensions of volume
Volume (V) is given by:
\[ V = \text{length}^3 = L^3 \]
Thus, the dimensions of volume are:
\[ [V] = [L^3] \]

### Step 4: Substitute the dimensions into the energy density formula
Now we can substitute the dimensions of energy and volume into the energy density formula:
\[ \text{Energy Density} = \frac{\text{Energy}}{\text{Volume}} = \frac{[M][L^2][T^{-2}]}{[L^3]} \]

### Step 5: Simplify the expression
When we simplify this expression, we get:
\[ \text{Energy Density} = [M][L^2][T^{-2}] \cdot [L^{-3}] = [M][L^{2-3}][T^{-2}] = [M][L^{-1}][T^{-2}] \]

### Final Result
Thus, the dimensions of energy density are:
\[ [E] = [M][L^{-1}][T^{-2}] \]

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