Which of the following combinations has the dimension of electrical resistance (`in_(0)` is the permittivity of vacuum and `mu_(0)` is the permeability of vacuum)?

Options

A. `sqrt((in_(0))/(mu_(0))`
B. `sqrt((mu_(0))/(in_(0)))`
C. `(mu_(0))/(in_(0))`
D. `(in_(0))/(mu_(0))`

Explanation

To determine which combination has the dimension of electrical resistance, we need to start by recalling the fundamental definition of electrical resistance (R).

### Step-by-Step Solution:

1. **Understanding Resistance**:
The electrical resistance \( R \) is defined as:
\[ R = \frac{V}{I} \]
where \( V \) is voltage and \( I \) is current.

2. **Expressing Voltage and Current**:
Voltage \( V \) can be expressed in terms of work done \( W \) and charge \( Q \):
\[ V = \frac{W}{Q} \]
Current \( I \) is defined as:
\[ I = \frac{Q}{T} \]
where \( T \) is time.

3. **Substituting for Current**:
Substituting the expression for current into the resistance formula, we have:
\[ R = \frac{W}{Q} \cdot \frac{T}{Q} = \frac{W \cdot T}{Q^2} \]

4. **Dimensions of Work and Charge**:
The dimension of work \( W \) is:
\[ [W] = [F \cdot d] = [M \cdot L^2 \cdot T^{-2}] \]
The dimension of charge \( Q \) is:
\[ [Q] = [I \cdot T] \]

5. **Substituting Dimensions**:
Now substituting the dimensions into the resistance formula:
\[ [R] = \frac{[M \cdot L^2 \cdot T^{-2}] \cdot [T]}{[I \cdot T]^2} = \frac{[M \cdot L^2 \cdot T^{-1}]}{[I^2 \cdot T^2]} = [M \cdot L^2 \cdot T^{-3} \cdot I^{-2}] \]

6. **Using Permittivity and Permeability**:
We know that:
\[ \epsilon_0 = \frac{1}{\mu_0 c^2} \]
where \( c \) is the speed of light. The dimensions of \( \epsilon_0 \) and \( \mu_0 \) can be derived as follows:
- The dimension of \( \epsilon_0 \) is:
\[ [\epsilon_0] = [M^{-1} \cdot L^{-3} \cdot T^4 \cdot I^2] \]
- The dimension of \( \mu_0 \) is:
\[ [\mu_0] = [M \cdot L \cdot T^{-2} \cdot I^{-2}] \]

7. **Finding the Combination**:
Now, we can find the combination \( \frac{\epsilon_0}{\mu_0} \):
\[ \frac{[\epsilon_0]}{[\mu_0]} = \frac{[M^{-1} \cdot L^{-3} \cdot T^4 \cdot I^2]}{[M \cdot L \cdot T^{-2} \cdot I^{-2}]} = [M^{-2} \cdot L^{-4} \cdot T^6 \cdot I^4] \]

8. **Final Comparison**:
The dimensions of electrical resistance derived earlier were:
\[ [R] = [M \cdot L^2 \cdot T^{-3} \cdot I^{-2}] \]
We need to find a combination that matches this dimension.

9. **Conclusion**:
After analyzing the combinations, we find that the combination that matches the dimension of electrical resistance is:
\[ \text{Option 2} \]