Which of the following dimensional formula is incorrect

Options

A. [permittivity of the free space `xx` electric field] `= M^(0)L^(-2)T^(1) A^(1)`]
B. [electrical conductivity] `= M^(-1) L^(-3) T^(3)A^(1)`]
C. [viscous force] `= [M^(1)L^(1)T^(-2)]`
D. `[("Electric field")/("Magnetic field")]=[M^(0)L^(1)T^(-1)]`

Answer

To determine which of the given dimensional formulas is incorrect, we will analyze each option step by step.

### Step 1: Analyze the first option - Permittivity of free space (ε₀) multiplied by electric field (E)

1. **Permittivity of free space (ε₀)** can be derived from Coulomb's law:
\[ ε₀ = \frac{Q^2}{F \cdot L^2} \]
where \(Q\) is charge, \(F\) is force, and \(L\) is distance.

2. **Electric field (E)** is defined as:
\[ E = \frac{F}{Q} \]

3. Combining these, we have:
\[ ε₀ \cdot E = \frac{Q^2}{F \cdot L^2} \cdot \frac{F}{Q} = \frac{Q}{L^2} \]

4. The dimensional formula for charge (Q) is \(A \cdot T\) (where A is ampere and T is time), and for distance (L) is \(L\). Thus:
\[ \text{Dimension of } ε₀ \cdot E = \frac{A \cdot T}{L^2} = M^0 L^{-2} A^1 T^1 \]

5. This matches the given option, so it is correct.

### Step 2: Analyze the second option - Electrical conductivity (σ)

1. **Electrical conductivity (σ)** is the reciprocal of resistivity (ρ):
\[ σ = \frac{1}{ρ} \]

2. Resistivity can be expressed as:
\[ ρ = R \cdot \frac{A}{L} \]
where \(R\) is resistance, \(A\) is area, and \(L\) is length.

3. The dimensional formula for resistance (R) can be derived from:
\[ R = \frac{V}{I} = \frac{M^1 L^2 T^{-3}}{A^1} \]
Thus, the dimension of resistance is:
\[ M^1 L^2 T^{-3} A^{-1} \]

4. Therefore, the dimension of resistivity (ρ) becomes:
\[ M^1 L^2 T^{-3} A^{-1} \cdot \frac{L}{A} = M^1 L^3 T^{-3} A^{-2} \]

5. Hence, the dimension of conductivity (σ) is:
\[ σ = \frac{1}{ρ} = M^{-1} L^{-3} T^{3} A^{2} \]

6. The option states \(M^{-1} L^{-3} T^{3} A^{1}\), which is incorrect since we derived \(A^{2}\) instead of \(A^{1}\).

### Step 3: Analyze the third option - Viscous force

1. **Viscous force** has the same dimensional formula as force:
\[ F = M \cdot L \cdot T^{-2} \]

2. Therefore, the dimension of viscous force is:
\[ M^1 L^1 T^{-2} \]

3. This matches the given option, so it is correct.

### Step 4: Analyze the fourth option - Electric field (E) divided by magnetic field (B)

1. **Electric field (E)** is defined as:
\[ E = \frac{F}{Q} \]

2. **Magnetic field (B)** can be derived from the Lorentz force law:
\[ B = \frac{F}{Q \cdot v} \]
where \(v\) is velocity.

3. Therefore, the dimension of electric field divided by magnetic field is:
\[ \frac{E}{B} = \frac{F/Q}{F/(Q \cdot v)} = v \]

4. The dimension of velocity is:
\[ L^1 T^{-1} \]

5. This matches the given option, so it is correct.

### Conclusion

After analyzing all options, the incorrect dimensional formula is for electrical conductivity (option b).