In terms of potential difference `C`, electric current`I` , permittivity `epsilon_(0)`, permeability `mu_(0)` and speed of light `c`, the dimensionally correct equation `(s)` is `(are)`

Options

A. (a) `mu_(0)I^(2) = epsilon_(0)V^(2)`
B. (b) `mu_(0)I = mu_(0)V`
C. ( c ) `I = epsilon_(0)cV`
D. (d) `mu_(0)cI = epsilon_(0)V`

Explanation

To determine the dimensionally correct equations involving potential difference \( V \), electric current \( I \), permittivity \( \epsilon_0 \), permeability \( \mu_0 \), and speed of light \( c \), we will analyze each option step by step.

### Step 1: Understand the relationships
We know from electromagnetic theory that:
1. The speed of light \( c \) is given by the equation:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
2. The resistance \( R \) can be expressed as:
\[ R = \frac{V}{I} \]

### Step 2: Analyze each option

#### Option A: \( \mu_0 I^2 = \epsilon_0 V^2 \)
Rearranging gives:
\[ \frac{\mu_0}{\epsilon_0} = \frac{V^2}{I^2} \]
Substituting \( R = \frac{V}{I} \) gives:
\[ \frac{\mu_0}{\epsilon_0} = R^2 \]
Since \( R \) has dimensions of \( \sqrt{\frac{\mu_0}{\epsilon_0}} \), this equation is dimensionally correct.

#### Option B: \( \mu_0 I = \mu_0 B \)
This option seems to have a typographical error, as it should involve \( \epsilon_0 \) instead of \( \mu_0 \). Therefore, we will consider it incorrect.

#### Option C: \( I = \epsilon_0 c V \)
Using \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \), we can rewrite:
\[ I = \epsilon_0 \cdot \frac{1}{\sqrt{\mu_0 \epsilon_0}} \cdot V \]
This simplifies to:
\[ I = \frac{V}{\sqrt{\mu_0/\epsilon_0}} \]
This is dimensionally correct since it relates current to voltage and the constants.

#### Option D: \( \mu_0 c I = \epsilon_0 V \)
Rearranging gives:
\[ \mu_0 c = \frac{\epsilon_0 V}{I} \]
Substituting \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \) gives:
\[ \mu_0 \cdot \frac{1}{\sqrt{\mu_0 \epsilon_0}} = \frac{\epsilon_0 V}{I} \]
This leads to:
\[ \sqrt{\frac{\mu_0}{\epsilon_0}} = \frac{\epsilon_0 V}{I} \]
This is not dimensionally consistent, thus it is incorrect.

### Conclusion
The dimensionally correct equations are:
- Option A: \( \mu_0 I^2 = \epsilon_0 V^2 \)
- Option C: \( I = \epsilon_0 c V \)