A current carrying circular loop is placed in an infinite plane if `phi `i is the magnetic flux through the inner region and `phi `o is magnitude of magnetic flux through the outer region, then

Options

A. `phi _i= - phi _ 0 `
B. ` phi_i= phi _ 0 `
C. `phi_ilt phi _ 0 `
D. `phi_igt phi_0 `

Answer

To solve the problem, we need to understand the relationship between the magnetic flux through the inner and outer regions of a current-carrying circular loop placed in an infinite plane.

### Step-by-Step Solution:

1. **Understanding Magnetic Flux**:
Magnetic flux (\( \Phi \)) through a surface is defined as the product of the magnetic field (\( B \)) passing through that surface and the area (\( A \)) of the surface, mathematically given by:
\[ \Phi = \int \vec{B} \cdot d\vec{A} \]
where \( d\vec{A} \) is the differential area vector.

2. **Magnetic Field Due to a Circular Loop**:
A circular loop carrying current generates a magnetic field that forms closed loops around the wire. The direction of the magnetic field inside the loop is different from that outside the loop.

3. **Analyzing the Inner and Outer Regions**:
- For the inner region of the loop, the magnetic field lines are directed inwards (towards the center of the loop).
- For the outer region, the magnetic field lines are directed outwards (away from the center of the loop).

4. **Direction of Magnetic Flux**:
Since the magnetic field direction is opposite in the inner and outer regions, the magnetic flux through these regions will also have opposite signs.

5. **Mathematical Representation**:
Let \( \Phi_i \) be the magnetic flux through the inner region and \( \Phi_o \) be the magnetic flux through the outer region. Due to the opposite directions of the magnetic fields in these regions, we can express the relationship as:
\[ \Phi_i = -\Phi_o \]
This indicates that the magnetic flux in the inner region is equal in magnitude but opposite in sign to the magnetic flux in the outer region.

6. **Conclusion**:
Therefore, the relationship between the magnetic fluxes through the inner and outer regions can be summarized as:
\[ \Phi_i = -\Phi_o \]

### Final Answer:
The relationship between the magnetic flux through the inner region (\( \Phi_i \)) and the outer region (\( \Phi_o \)) is:
\[ \Phi_i = -\Phi_o \]