Let `epsi_0` denote the dimensional formula of the permittivity of vacuum. If M= mass , L=length , T=time and A=electric current , then dimension of permittivity is given as `[M^(p)L^(q)T^(r)A^(s)]`. Find the value of `(p-q+r)/(s)`

Answer

To solve the problem, we need to find the dimensional formula of the permittivity of vacuum, denoted as \( \epsilon_0 \). We will use Coulomb's law as a basis for our calculations.

### Step 1: Understand Coulomb's Law
Coulomb's law states that the force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
From this, we can express \( \epsilon_0 \) as:
\[ \epsilon_0 = \frac{q_1 q_2}{F r^2} \cdot 4\pi \]

### Step 2: Determine the Dimensions
We need to find the dimensions of \( \epsilon_0 \).

1. **Dimensions of Charge \( q \)**:
- Charge \( q \) can be expressed in terms of current \( I \) and time \( T \):
\[ q = I \cdot T \]
Therefore, the dimension of charge \( q \) is:
\[ [q] = [I][T] = A \cdot T \]

2. **Dimensions of Force \( F \)**:
- The dimension of force is given by Newton's second law \( F = m \cdot a \):
\[ [F] = [M][L][T^{-2}] = M \cdot L \cdot T^{-2} \]

3. **Dimensions of Distance \( r \)**:
- The dimension of distance \( r \) is simply:
\[ [r] = [L] \]

### Step 3: Substitute Dimensions into the Formula
Now, substituting the dimensions into the expression for \( \epsilon_0 \):
\[ \epsilon_0 = \frac{(q_1)(q_2)}{F r^2} \]
Substituting the dimensions we have:
\[ [\epsilon_0] = \frac{(A \cdot T)(A \cdot T)}{M \cdot L \cdot T^{-2} \cdot L^2} \]
This simplifies to:
\[ [\epsilon_0] = \frac{A^2 \cdot T^2}{M \cdot L^3 \cdot T^{-2}} = \frac{A^2 \cdot T^2 \cdot T^2}{M \cdot L^3} = \frac{A^2 \cdot T^4}{M \cdot L^3} \]

### Step 4: Write the Dimensional Formula
Thus, the dimensional formula of \( \epsilon_0 \) is:
\[ [\epsilon_0] = M^{-1} L^{-3} T^4 A^2 \]

### Step 5: Identify \( p, q, r, s \)
From the dimensional formula \( [M^p L^q T^r A^s] \):
- \( p = -1 \)
- \( q = -3 \)
- \( r = 4 \)
- \( s = 2 \)

### Step 6: Calculate \( \frac{p - q + r}{s} \)
Now, we can calculate:
\[ \frac{p - q + r}{s} = \frac{-1 - (-3) + 4}{2} = \frac{-1 + 3 + 4}{2} = \frac{6}{2} = 3 \]

### Final Answer
Thus, the value of \( \frac{p - q + r}{s} \) is:
\[ \boxed{3} \]