A circular loop of radius `R` , carrying `I`, lies in `x- y` plane with its origin . The total magnetic flux through ` x-y` plane is

Options

A. Directly proportional to R
B. Directly proportional to I
C. Inversely proportional to I
D. Zero

Answer

To find the total magnetic flux through the x-y plane due to a circular loop of radius \( R \) carrying a current \( I \), we can follow these steps:

### Step 1: Understand the Magnetic Field due to the Circular Loop
A circular loop carrying a current generates a magnetic field around it. The direction of the magnetic field can be determined using the right-hand rule. For a loop lying in the x-y plane, the magnetic field lines will emerge from the loop's center and form closed loops around the wire.

### Step 2: Analyze the Magnetic Field Lines
The magnetic field lines created by the current in the loop will enter the x-y plane at some points and exit at others, forming closed loops. This means that for every magnetic field line entering the x-y plane, there is a corresponding line exiting the plane.

### Step 3: Calculate the Magnetic Flux
Magnetic flux (\( \Phi_B \)) through a surface is given by the formula:
\[ \Phi_B = \int \vec{B} \cdot d\vec{A} \]
where \( \vec{B} \) is the magnetic field and \( d\vec{A} \) is the differential area vector.

### Step 4: Consider the Orientation of the Area Vector
In this case, the area vector \( d\vec{A} \) for the x-y plane points in the z-direction. However, the magnetic field lines due to the loop are oriented in such a way that they enter and exit the x-y plane, resulting in equal contributions of magnetic field entering and exiting the plane.

### Step 5: Conclude the Total Magnetic Flux
Since the magnetic field lines that enter the x-y plane are equal in magnitude and opposite in direction to those that exit, the total magnetic flux through the x-y plane sums to zero. Therefore, the total magnetic flux through the x-y plane is:
\[ \Phi_B = 0 \]

### Final Answer
The total magnetic flux through the x-y plane is \( 0 \).

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