`[("Permeability")/("Permittivity")]` will have the dimensions of

Options

A. `M^(@)L^(@)T^(@)A^(@)`
B. `M^(2)L^(2)T^(4)A^(2)`
C. `M^(2)L^(4)T^(-6)A^(-4)`
D. `M^(-2)L^(-4)T^(6)A^(4)`

Answer

To solve the problem of finding the dimensions of the ratio of permeability to permittivity, we will follow these steps:

### Step 1: Determine the dimensions of permeability (μ₀)
Permeability (μ₀) is defined in terms of the magnetic force between two parallel current-carrying conductors. The formula for the magnetic force per unit length (F/L) between two conductors is given by:

\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi D} \]

From this, we can express μ₀ as:

\[ \mu_0 = \frac{F \cdot 2\pi D}{L \cdot I_1 I_2} \]

Now, substituting the dimensions:
- Force (F) has dimensions of \(MLT^{-2}\)
- Length (L) has dimensions of \(L\)
- Current (I) has dimensions of \(A\)

Thus, the dimensions of μ₀ can be calculated as follows:

\[ \text{Dimensions of } \mu_0 = \frac{MLT^{-2} \cdot L}{A^2} = ML T^{-2} A^{-2} \]

### Step 2: Determine the dimensions of permittivity (ε₀)
Permittivity (ε₀) can be defined using the electrostatic force between two point charges. The formula for the electrostatic force (F) between two charges is given by:

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

Rearranging gives:

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

Substituting the dimensions:
- Charge (Q) has dimensions of \(AT\) (Ampere × Time)
- Force (F) has dimensions of \(MLT^{-2}\)
- Distance (D) has dimensions of \(L\)

Thus, the dimensions of ε₀ can be calculated as follows:

\[ \text{Dimensions of } \epsilon_0 = \frac{1}{4\pi} \cdot \frac{(AT)^2}{MLT^{-2} \cdot L^2} = M^{-1} L^{-3} T^4 A^2 \]

### Step 3: Calculate the dimensions of the ratio (μ₀/ε₀)
Now we can find the dimensions of the ratio of permeability to permittivity:

\[ \frac{\mu_0}{\epsilon_0} = \frac{MLT^{-2} A^{-2}}{M^{-1} L^{-3} T^4 A^2} \]

This simplifies to:

\[ = \frac{M^2 L^4 T^{-6} A^{-4}} \]

### Step 4: Conclusion
Thus, the dimensions of the ratio of permeability to permittivity are:

\[ M^2 L^4 T^{-6} A^{-4} \]

### Final Answer
The correct option is C: \(M^2 L^4 T^{-6} A^{-4}\).

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