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 gt phi o`
B. `phi I lt phi o`
C. `phi I = - phi o`
D. `phi I = phi o`

Answer

To solve the problem, we need to analyze 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 the Setup**:
- We have a circular loop that carries a current. Let's assume the current flows in a clockwise direction.
- The magnetic field generated by the loop will have specific characteristics both inside the loop and outside of it.

2. **Magnetic Field Inside the Loop**:
- Inside the loop, the magnetic field lines are directed into the plane of the loop. This is represented by "cross" symbols (×).
- The strength of the magnetic field (B) inside the loop can be calculated using Ampere's Law or Biot-Savart Law, but for our purpose, we just need to know that it is directed into the loop.

3. **Magnetic Field Outside the Loop**:
- Outside the loop, the magnetic field lines are directed out of the plane. This is represented by "dot" symbols (•).
- The magnetic field outside the loop decreases with distance from the loop, but for an infinite plane, we can assume it extends infinitely.

4. **Magnetic Flux Calculation**:
- Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) through which it passes:
\[ \Phi = B \cdot A \]
- For the inner region (Φi), the magnetic flux is:
\[ \Phi_i = B_{\text{inside}} \cdot A_{\text{inner}} \]
- For the outer region (Φo), the magnetic flux is:
\[ \Phi_o = B_{\text{outside}} \cdot A_{\text{outer}} \]

5. **Relation Between Φi and Φo**:
- The number of magnetic field lines entering the loop (inside) is equal to the number of lines exiting the loop (outside). However, since they are in opposite directions, we can establish a relationship:
\[ \Phi_i = -\Phi_o \]
- This indicates that the magnetic flux through the inner region is equal in magnitude but opposite in direction to the magnetic flux through the outer region.

### Final Answer:
\[ \Phi_i = -\Phi_o \]