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

Options

A. `mu_(0)I^(2) = epsilon V^(2)`
B. `mu_(0) I = mu_(0) V`
C. `I = epsilon_(0) V`
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: Identify the dimensions of each quantity

1. **Potential Difference \( V \)**:
- Dimension: \( [V] = [ML^2T^{-3}A^{-1}] \)

2. **Electric Current \( I \)**:
- Dimension: \( [I] = [A] \)

3. **Permittivity \( \epsilon_0 \)**:
- Dimension: \( [\epsilon_0] = [M^{-1}L^{-3}T^4A^2] \)

4. **Permeability \( \mu_0 \)**:
- Dimension: \( [\mu_0] = [MLT^{-2}A^{-2}] \)

5. **Speed of Light \( c \)**:
- Dimension: \( [c] = [LT^{-1}] \)

### Step 2: Analyze each option for dimensional consistency

#### Option 1: \( \mu_0 I^2 = \epsilon_0 V^2 \)

- Left-hand side:
\[ [\mu_0 I^2] = [MLT^{-2}A^{-2}] \cdot [A^2] = [MLT^{-2}] \]

- Right-hand side:
\[ [\epsilon_0 V^2] = [M^{-1}L^{-3}T^4A^2] \cdot [ML^2T^{-3}A^{-1}]^2 = [M^{-1}L^{-3}T^4A^2] \cdot [M^2L^4T^{-6}A^{-2}] \]
\[ = [M^{1}L^{1}T^{-2}] \]

- Both sides:
\[ [MLT^{-2}] = [MLT^{-2}] \]
- Conclusion: **Option 1 is correct.**

#### Option 2: \( \mu_0 I = \mu_0 V \)

- Cancel \( \mu_0 \):
\[ [I] = [V] \]
- This implies:
\[ [A] = [ML^2T^{-3}A^{-1}] \]
- Conclusion: **Option 2 is incorrect.**

#### Option 3: \( I = \epsilon_0 V \)

- Left-hand side:
\[ [I] = [A] \]

- Right-hand side:
\[ [\epsilon_0 V] = [M^{-1}L^{-3}T^4A^2] \cdot [ML^2T^{-3}A^{-1}] = [M^{0}L^{-1}T^{1}A^{1}] \]
- This implies:
\[ [A] \neq [M^{0}L^{-1}T^{1}A^{1}] \]
- Conclusion: **Option 3 is incorrect.**

#### Option 4: \( \mu_0 c I = \epsilon_0 V \)

- Left-hand side:
\[ [\mu_0 c I] = [MLT^{-2}A^{-2}] \cdot [LT^{-1}] \cdot [A] = [M L^2 T^{-3}] \]

- Right-hand side:
\[ [\epsilon_0 V] = [M^{-1}L^{-3}T^4A^2] \cdot [ML^2T^{-3}] = [M^{0}L^{-1}T^{1}A^{1}] \]
- This implies:
\[ [M L^2 T^{-3}] \neq [M^{0}L^{-1}T^{1}A^{1}] \]
- Conclusion: **Option 4 is incorrect.**

### Final Conclusion
The only dimensionally correct equation is **Option 1: \( \mu_0 I^2 = \epsilon_0 V^2 \)**.

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