Using mass (M), length (L), time (T) and current (A) as fundamental quantities, the dimension of permittivity is:

Options

A. `ML^(-2)T^(2)A`
B. `M^(-1)L^(-3)T^(4)A2^(2)`
C. `MLT^(-2)A`
D. `ML^(2)T^(-1)A^(2)`

Explanation

To find the dimension of permittivity using mass (M), length (L), time (T), and current (A) as fundamental quantities, we start with the formula for the force between two charged particles:

\[ F = \frac{q_1 q_2}{4 \pi \epsilon_0 r^2} \]

Where:
- \( F \) is the force,
- \( q_1 \) and \( q_2 \) are the charges,
- \( \epsilon_0 \) is the permittivity of free space,
- \( r \) is the distance between the charges.

### Step 1: Rearranging the Formula
We can rearrange the formula to express permittivity (\( \epsilon_0 \)):

\[ \epsilon_0 = \frac{q_1 q_2}{4 \pi F r^2} \]

### Step 2: Identifying Dimensions
Next, we need to identify the dimensions of each component in the equation.

1. **Force (F)**: The dimension of force is given by:
\[ [F] = [M][L][T^{-2}] \]

2. **Charge (q)**: The dimension of charge can be expressed in terms of current and time:
\[ [q] = [A][T] \]
Therefore, for \( q_1 \) and \( q_2 \):
\[ [q_1] = [A][T], \quad [q_2] = [A][T] \]
Thus,
\[ [q_1 q_2] = [A^2][T^2] \]

3. **Distance (r)**: The dimension of distance is:
\[ [r] = [L] \]
Therefore,
\[ [r^2] = [L^2] \]

### Step 3: Substituting Dimensions into the Formula
Now we substitute these dimensions back into the equation for permittivity:

\[ [\epsilon_0] = \frac{[A^2][T^2]}{[M][L][T^{-2}][L^2]} \]

### Step 4: Simplifying the Expression
Now we simplify the expression:

\[ [\epsilon_0] = \frac{[A^2][T^2]}{[M][L^3][T^{-2}]} \]

This can be rewritten as:

\[ [\epsilon_0] = \frac{[A^2][T^4]}{[M][L^3]} \]

### Step 5: Final Form
Rearranging gives us the final dimension of permittivity:

\[ [\epsilon_0] = [M^{-1}][L^{-3}][T^4][A^2] \]

### Conclusion
Thus, the dimension of permittivity in terms of fundamental quantities is:

\[ [\epsilon_0] = M^{-1} L^{-3} T^4 A^2 \]