Obtain the dimensional formula of `epsilon_(0)`.

Answer

To find the dimensional formula of \( \epsilon_0 \) (the permittivity of free space), we can start from the formula of Coulomb's law, which relates the force between two point charges to their magnitudes and the distance between them.

### Step-by-Step Solution:

1. **Coulomb's Law**: The formula for the electrostatic force \( F \) between two charges \( Q_1 \) and \( Q_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{K Q_1 Q_2}{r^2} \]
where \( K \) is the Coulomb's constant.

2. **Coulomb's Constant**: The Coulomb's constant \( K \) can also be expressed in terms of \( \epsilon_0 \):
\[ K = \frac{1}{4 \pi \epsilon_0} \]
Substituting this into the equation gives:
\[ F = \frac{Q_1 Q_2}{4 \pi \epsilon_0 r^2} \]

3. **Rearranging for \( \epsilon_0 \)**: Rearranging the equation to solve for \( \epsilon_0 \):
\[ \epsilon_0 = \frac{Q_1 Q_2}{4 \pi F r^2} \]

4. **Identifying Dimensions**: Now, we need to find the dimensions of each term in the equation:
- The dimension of force \( F \) is:
\[ [F] = M L T^{-2} \]
- The dimension of charge \( Q \) can be expressed as:
\[ [Q] = I T \]
where \( I \) is the current in amperes and \( T \) is time in seconds.

5. **Substituting Dimensions**: Substituting the dimensions into the equation for \( \epsilon_0 \):
\[ [\epsilon_0] = \frac{[Q]^2}{[F] [r]^2} = \frac{(I T)^2}{(M L T^{-2}) (L^2)} \]

6. **Calculating Dimensions**: Now we can calculate the dimensions:
- The numerator becomes:
\[ (I T)^2 = I^2 T^2 \]
- The denominator becomes:
\[ M L T^{-2} \cdot L^2 = M L^3 T^{-2} \]
- Therefore, we have:
\[ [\epsilon_0] = \frac{I^2 T^2}{M L^3 T^{-2}} = \frac{I^2 T^4}{M L^3} \]

7. **Final Dimensional Formula**: Thus, the dimensional formula for \( \epsilon_0 \) is:
\[ [\epsilon_0] = M^{-1} L^{-3} T^{4} I^{2} \]