What is the dimensional formula of electric energy density ?

Options

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

Explanation

To find the dimensional formula of electric energy density, we can follow these steps:

### Step 1: Understand the formula for electric energy density
The electric energy density \( u \) is given by the formula:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
where \( \epsilon_0 \) is the permittivity of free space and \( E \) is the electric field.

### Step 2: Identify the dimensional formula of \( \epsilon_0 \)
To find the dimensional formula of \( \epsilon_0 \), we can use the relation from Coulomb's law:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
Rearranging gives:
\[ \epsilon_0 = \frac{q_1 q_2}{F r^2} \]

### Step 3: Write the dimensional formulas
- The dimensional formula of charge \( q \) is \( [Q] \) or \( [A T] \) (where \( A \) is the unit of current and \( T \) is time).
- The dimensional formula of force \( F \) is \( [M L T^{-2}] \).
- The dimensional formula of distance \( r \) is \( [L] \).

Substituting these into the expression for \( \epsilon_0 \):
\[ [\epsilon_0] = \frac{[Q]^2}{[F][L^2]} = \frac{(A T)^2}{(M L T^{-2}) L^2} = \frac{A^2 T^2}{M L^3 T^{-2}} = \frac{A^2 T^4}{M L^3} \]
Thus, the dimensional formula of \( \epsilon_0 \) is:
\[ [\epsilon_0] = M^{-1} L^{-3} A^2 T^4 \]

### Step 4: Identify the dimensional formula of electric field \( E \)
From the relation \( F = qE \), we can express \( E \) as:
\[ E = \frac{F}{q} \]
Substituting the dimensional formulas:
\[ [E] = \frac{[F]}{[Q]} = \frac{[M L T^{-2}]}{[A T]} = \frac{M L T^{-2}}{A T} = \frac{M L}{A T^2} \]

### Step 5: Calculate the dimensional formula of \( E^2 \)
Now, we need to square the dimensional formula of \( E \):
\[ [E^2] = \left(\frac{M L}{A T^2}\right)^2 = \frac{M^2 L^2}{A^2 T^4} \]

### Step 6: Substitute into the formula for \( u \)
Now substitute the dimensional formulas of \( \epsilon_0 \) and \( E^2 \) into the energy density formula:
\[ [u] = [\epsilon_0][E^2] = \left(M^{-1} L^{-3} A^2 T^4\right) \left(\frac{M^2 L^2}{A^2 T^4}\right) \]
This simplifies to:
\[ [u] = M^{-1} L^{-3} A^2 T^4 \cdot \frac{M^2 L^2}{A^2 T^4} = M^{1} L^{-1} T^{0} A^{0} \]

### Step 7: Final dimensional formula
Thus, the dimensional formula of electric energy density \( u \) is:
\[ [u] = M^1 L^{-1} T^{-2} \]

### Summary
The dimensional formula of electric energy density is:
\[ M^1 L^{-1} T^{-2} \]