A circular coil, carrying a constant current i s kept in the x - y plane. The magnetic flux through the entire x - y plane exluding the area of the circular coil is given by `phi` and the magetic flux through the area of the circular coil area is given by `phi_(0)`, then

Options

A. `phi gt -phi_(0)`
B. `phi lt phi_(0)`
C. `phi = -phi_(0)`
D. `phi = phi_(0)`

Answer

To solve the problem, we need to analyze the magnetic flux through a circular coil and the surrounding area in the x-y plane. Here's a step-by-step breakdown of the solution:

### Step 1: Understanding Magnetic Flux
Magnetic flux (Φ) through a surface is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where \(B\) is the magnetic field, \(A\) is the area, and \(\theta\) is the angle between the magnetic field and the normal to the surface.

### Step 2: Magnetic Field Due to the Circular Coil
For a circular coil carrying a constant current \(I\), the magnetic field \(B\) at a point in the plane of the coil can be expressed as:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where \(r\) is the distance from the center of the coil to the point where the magnetic field is being calculated, and \(\mu_0\) is the permeability of free space.

### Step 3: Magnetic Flux Through the Coil
Let’s denote the magnetic flux through the area of the circular coil as \(\Phi_0\). The area \(A_0\) of the circular coil is given by:
\[ A_0 = \pi R^2 \]
where \(R\) is the radius of the circular coil. Thus, the magnetic flux through the coil can be written as:
\[ \Phi_0 = B \cdot A_0 \cdot \cos(0) = B \cdot A_0 \]
Since \(\cos(0) = 1\), we have:
\[ \Phi_0 = \left(\frac{\mu_0 I}{2\pi R}\right) \cdot (\pi R^2) = \frac{\mu_0 I R}{2} \]

### Step 4: Magnetic Flux Through the Surrounding Area
Let’s denote the magnetic flux through the entire x-y plane excluding the area of the circular coil as \(\Phi\). The magnetic field in the surrounding area is still influenced by the current in the coil, but the direction of the magnetic field lines will be opposite to that within the coil. Therefore, the magnetic flux through the surrounding area can be expressed similarly:
\[ \Phi = -\Phi_0 \]
This negative sign indicates that the direction of the magnetic flux through the surrounding area is opposite to that through the coil.

### Step 5: Conclusion
From the above analysis, we can conclude that:
\[ \Phi = -\Phi_0 \]

### Final Answer
Thus, the relationship between the magnetic flux through the entire x-y plane excluding the area of the circular coil (\(\Phi\)) and the magnetic flux through the area of the circular coil (\(\Phi_0\)) is:
\[ \Phi = -\Phi_0 \]