A straight section PQ of a circuit lies along the x-axis from `x=-(a)/(2)` to `x=(a)/(2)` and carries a steady current i. The magnetic field due to the section PQ at a point x =+a will be

Options

A. proportional to a
B. proportional to `a^(2)`
C. proportional to 1/a
D. zero

Answer

To solve the problem of finding the magnetic field due to a straight section PQ of a circuit at a point x = +a, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- The straight section PQ lies along the x-axis from \( x = -\frac{a}{2} \) to \( x = \frac{a}{2} \).
- The point where we need to find the magnetic field is at \( x = a \).

2. **Using Biot-Savart Law**:
- The Biot-Savart Law states that the magnetic field \( \mathbf{B} \) at a point due to a current-carrying conductor is given by:
\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} \]
- Here, \( I \) is the current, \( d\mathbf{l} \) is the current element, and \( \mathbf{r} \) is the position vector from the current element to the point where the field is being calculated.

3. **Identifying the Geometry**:
- For the straight section PQ, we can consider a small current element \( d\mathbf{l} \) at a position \( x \) (where \( -\frac{a}{2} \leq x \leq \frac{a}{2} \)).
- The position vector \( \mathbf{r} \) from the current element to the point \( x = a \) is \( \mathbf{r} = a - x \).

4. **Calculating the Angle**:
- The angle \( \theta \) between the current element \( d\mathbf{l} \) and the vector \( \mathbf{r} \) is crucial for applying the Biot-Savart Law.
- Since the current flows along the x-axis and the point is also on the x-axis, the angle \( \theta \) between \( d\mathbf{l} \) and \( \mathbf{r} \) is \( 0^\circ \).

5. **Evaluating the Magnetic Field**:
- Since \( \sin(0^\circ) = 0 \), the contribution to the magnetic field from each current element \( d\mathbf{l} \) becomes zero:
\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} = 0 \]
- Therefore, the total magnetic field \( \mathbf{B} \) at point \( x = a \) is also zero.

6. **Conclusion**:
- The magnetic field due to the straight section PQ at point \( x = a \) is:
\[ \mathbf{B} = 0 \]

### Final Answer:
The magnetic field at point \( x = a \) is \( 0 \). Therefore, the correct option is option 4.