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

Options

A. `M^(-1)LT^(-2)A`
B. `ML^2T(-2)A^(-1)`
C. `MLT^(-2)A^(-2)`
D. `MLT(-1)A^(-1)`

Explanation

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

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

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 to isolate permittivity
From the equation, we can express permittivity \( \epsilon_0 \) as:

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

### Step 2: Identifying dimensions of each quantity
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 (I) and time (T):
\[ [q] = A T \]
Therefore, for two charges \( q_1 \) and \( q_2 \):
\[ [q_1 q_2] = (A T)(A T) = A^2 T^2 \]

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

### Step 3: Substituting dimensions back into the permittivity equation
Now, substituting these dimensions into the equation for \( \epsilon_0 \):

\[ [\epsilon_0] = \frac{[q_1 q_2]}{[F][r^2]} = \frac{A^2 T^2}{(M L T^{-2})(L^2)} \]

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

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

This can be rewritten as:

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

### Final Result
Thus, the dimension of permittivity \( \epsilon_0 \) in terms of fundamental quantities is:

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

### Summary
The dimension of permittivity is:

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