Let `[epsi_(0)]` denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T= time and A= electric current, then

Options

A. `[epsi_(0)]=[M^(-1)L^(2)T^(-1)A]`
B. `[epsi_(0)]=[M^(-1)L^(-3)T^(2)A]`
C. `[epsi_(0)]=[M^(-1)L^(-3)T^(4)A^(2)]`
D. `[epsi_(0)]=[M^(-1)L^(2)T^(-1)A^(-2)]`

Explanation

To find the dimensional formula of the permittivity of vacuum (denoted as \(\epsilon_0\)), we can use Coulomb's law, which relates the force between two point charges to the distance between them. Here’s a step-by-step solution:

### Step 1: Write down Coulomb's Law
According to Coulomb's law, the force \(F\) between two charges \(Q_1\) and \(Q_2\) separated by a distance \(r\) is given by:

\[ F = \frac{1}{4\pi \epsilon_0} \frac{Q_1 Q_2}{r^2} \]

### Step 2: Rearrange the equation to find \(\epsilon_0\)
From the equation, we can rearrange it to express \(\epsilon_0\):

\[ \epsilon_0 = \frac{1}{4\pi} \frac{Q_1 Q_2}{F r^2} \]

### Step 3: Substitute the dimensions of the quantities
- The dimension of force \(F\) is given by:

\[ [F] = M L T^{-2} \]

- The charge \(Q\) can be expressed in terms of current \(I\) and time \(t\):

\[ Q = I \cdot t \]

Thus, the dimension of charge \(Q\) is:

\[ [Q] = A T \]

### Step 4: Substitute \(Q_1\) and \(Q_2\) into the equation
Substituting \(Q_1\) and \(Q_2\) into the equation for \(\epsilon_0\):

\[ Q_1 Q_2 = (A T)(A T) = A^2 T^2 \]

### Step 5: Substitute everything back into the equation for \(\epsilon_0\)
Now substituting the dimensions back into the equation for \(\epsilon_0\):

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

### Step 6: Simplify the expression
This simplifies to:

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

### Step 7: Write the final dimensional formula
Thus, the dimensional formula for the permittivity of vacuum \(\epsilon_0\) is:

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

### Conclusion
The dimensional formula of the permittivity of vacuum is:

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