A circular coil is in `y-z` plane with centre at origin. The coil is carrying a constant current. Assuming direction of magnetic field at `x = – 25 cm` to be positive direction of magnetic field, which of the following graphs shows variation of magnetic field along x-axis

Options

A.
B.
C.
D.

Answer

To solve the problem step by step, we need to analyze the situation described and derive the relationship between the magnetic field and the position along the x-axis.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A circular coil is placed in the y-z plane with its center at the origin (0, 0, 0).
- The coil carries a constant current, which generates a magnetic field around it.

2. **Direction of the Magnetic Field**:
- The problem states that the direction of the magnetic field at \( x = -25 \, \text{cm} \) is considered to be in the positive direction.
- Due to the symmetry of the circular coil, the magnetic field will point in the same direction on both sides of the coil (i.e., for \( x < 0 \) and \( x > 0 \)).

3. **Magnetic Field Formula**:
- The magnetic field \( B \) at a distance \( r \) from a circular coil can be expressed using the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3} \]
- Here, \( m \) is the magnetic dipole moment, which can be calculated as:
\[ m = n \cdot I \cdot A \]
where \( n \) is the number of turns, \( I \) is the current, and \( A \) is the area of the coil.

4. **Substituting for \( r \)**:
- In our case, we can replace \( r \) with \( x \) (the distance along the x-axis), leading to:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{n \cdot I \cdot A}{x^3} \]

5. **Behavior of Magnetic Field**:
- From the equation \( B \propto \frac{1}{x^3} \), we can deduce that:
- As \( x \) approaches 0 (the coil center), \( B \) becomes very large (theoretically infinite).
- As \( |x| \) increases (moving away from the coil), \( B \) decreases rapidly.

6. **Graphical Representation**:
- The magnetic field \( B \) is positive for both \( x < 0 \) and \( x > 0 \) due to the symmetry and the direction of the magnetic field.
- The graph of \( B \) versus \( x \) will show that \( B \) approaches infinity as \( x \) approaches 0 and decreases as \( |x| \) increases.

7. **Identifying the Correct Graph**:
- Based on the above analysis, the correct graph will show:
- A positive magnetic field for both negative and positive values of \( x \).
- A steep drop in the magnetic field as we move away from the center of the coil.

### Conclusion:
The correct graph that represents the variation of the magnetic field along the x-axis is the one that shows a positive value for \( B \) on both sides of the origin and a rapid decrease as \( |x| \) increases.