Keeping current per unit length of arc constant, the variation of magnetic field at the centre of an arc (B) with the angle subtended by the arc at the centre `(theta)` can be best represented by

Options

A.
B.
C.
D.

Answer

To solve the problem of how the magnetic field \( B \) at the center of an arc varies with the angle \( \theta \) subtended by the arc at the center, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Magnetic Field**:
The magnetic field \( B \) at the center of an arc is given by the formula:
\[ B = \frac{\mu_0 I}{2r} \cdot \frac{\theta}{2\pi} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the arc,
- \( r \) is the radius of the arc,
- \( \theta \) is the angle subtended by the arc at the center.

2. **Relate Current to Arc Length**:
The length \( L \) of the arc can be expressed as:
\[ L = r\theta \]
Given that the current per unit length of the arc is constant, we can denote this constant as \( k \). Thus, we have:
\[ \frac{I}{L} = k \implies I = kL = k(r\theta) \]

3. **Substitute Current into the Magnetic Field Formula**:
Now, substitute \( I = k(r\theta) \) into the magnetic field formula:
\[ B = \frac{\mu_0 (k r \theta)}{2r} \cdot \frac{\theta}{2\pi} \]

4. **Simplify the Expression**:
Simplifying the expression:
\[ B = \frac{\mu_0 k \theta}{2} \cdot \frac{\theta}{2\pi} = \frac{\mu_0 k \theta^2}{4\pi} \]

5. **Identify the Relationship**:
From the final expression, we see that:
\[ B \propto \theta^2 \]
This indicates that the magnetic field \( B \) varies with the square of the angle \( \theta \), which is a parabolic relationship.

6. **Conclusion**:
Therefore, the variation of the magnetic field \( B \) at the center of the arc with the angle \( \theta \) can be best represented as a parabolic function.

### Final Answer:
The correct option is that \( B \) varies as \( \theta^2 \), which corresponds to a parabolic function.

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