Consider a rectangular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux through an infinite plane that contains the rectangular coil and excluding the rectangular area is given by`phi`. magnetic flux through the area of the rectangular area is given by `phi`. Which of the following option is correct?

Options

A. `phi_(i)=phi_(0)`
B. `phi_(i)=-phi_(0)`
C. `phi_(i)gtphi_(0)`
D. `phi_(i)ltphi_(0)`

Explanation

To solve the problem, we need to analyze the magnetic flux through a rectangular coil of wire carrying a constant current \( I \) and how it relates to the magnetic flux through an infinite plane that contains the coil.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a rectangular coil carrying a current \( I \). This coil generates a magnetic field around it.
- We denote the magnetic flux through the area of the rectangular coil as \( \Phi_0 \) and the magnetic flux through the infinite plane (excluding the area of the rectangular coil) as \( \Phi_i \).

2. **Magnetic Field Inside and Outside the Coil**:
- The magnetic field inside the coil, denoted as \( B_0 \), is stronger than the magnetic field outside the coil, denoted as \( B_i \). This can be understood using the right-hand thumb rule, which indicates the direction of the magnetic field lines generated by the current in the coil.
- Thus, we establish the relationship:
\[ B_0 > B_i \]

3. **Calculating Magnetic Flux**:
- The magnetic flux \( \Phi \) through an area is given by the formula:
\[ \Phi = B \cdot A \]
- For the rectangular coil, the flux is:
\[ \Phi_0 = B_0 \cdot A \]
- For the infinite plane (excluding the area of the rectangular coil), the flux is:
\[ \Phi_i = B_i \cdot A \]

4. **Comparing the Fluxes**:
- Since we established that \( B_0 > B_i \), it follows that:
\[ \Phi_0 = B_0 \cdot A > B_i \cdot A = \Phi_i \]
- Therefore, we conclude that:
\[ \Phi_0 > \Phi_i \]

5. **Final Conclusion**:
- The magnetic flux through the area of the rectangular coil is greater than the magnetic flux through the infinite plane that excludes the rectangular area. Thus, the correct option is:
\[ \Phi_0 > \Phi_i \]