Dimension of resistance `R` in terms of `mu0 & epsilon_(0)` are

Options

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

Answer

To find the dimension of resistance \( R \) in terms of the permeability \( \mu_0 \) and permittivity \( \epsilon_0 \), we will follow these steps:

### Step 1: Write the dimensions of \( \mu_0 \) and \( \epsilon_0 \)

The permeability \( \mu_0 \) has the dimension:
\[ [\mu_0] = M L T^{-2} A^{-2} \]

The permittivity \( \epsilon_0 \) has the dimension:
\[ [\epsilon_0] = M^{-1} L^{-3} T^{-4} A^{2} \]

### Step 2: Find the dimension of \( \frac{\mu_0}{\epsilon_0} \)

Now, we will calculate the dimension of \( \frac{\mu_0}{\epsilon_0} \):
\[ \frac{\mu_0}{\epsilon_0} = \frac{M L T^{-2} A^{-2}}{M^{-1} L^{-3} T^{-4} A^{2}} \]

### Step 3: Simplify the expression

When we divide the dimensions, we subtract the exponents of like bases:
\[ = M^{1 - (-1)} L^{1 - (-3)} T^{-2 - (-4)} A^{-2 - 2} \]
This simplifies to:
\[ = M^{2} L^{4} T^{2} A^{-4} \]

### Step 4: Write the dimension of resistance \( R \)

The dimension of resistance \( R \) is given by:
\[ [R] = M^{1} L^{2} T^{-3} A^{-2} \]

### Step 5: Relate \( R \) to \( \frac{\mu_0}{\epsilon_0} \)

To express \( R \) in terms of \( \frac{\mu_0}{\epsilon_0} \), we can observe that:
\[ R \propto \sqrt{\frac{\mu_0}{\epsilon_0}} \]

### Conclusion

Thus, we conclude that the dimension of resistance \( R \) in terms of \( \mu_0 \) and \( \epsilon_0 \) is:
\[ R = \sqrt{\frac{\mu_0}{\epsilon_0}} \]