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 find the magnetic field at point \( x = +a \) due to the straight section \( PQ \) of the circuit carrying a steady current \( i \), we will use the Biot-Savart Law. Here's a step-by-step solution:

### Step 1: Define the Geometry of the Problem
The section \( PQ \) of the circuit lies along the x-axis from \( x = -\frac{a}{2} \) to \( x = \frac{a}{2} \). The current \( i \) flows through this section. We need to find the magnetic field at the point \( x = +a \).

**Hint:** Visualize the setup by sketching the x-axis with points P and Q marked, and indicate the direction of current.

### Step 2: Apply the Biot-Savart Law
The Biot-Savart Law states that the magnetic field \( dB \) due to a current element \( i \, dl \) at a distance \( r \) is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{i \, dl \, \sin \theta}{r^2} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( \theta \) is the angle between the current element \( dl \) and the position vector \( r \),
- \( r \) is the distance from the current element to the point where the magnetic field is being calculated.

**Hint:** Remember that \( \sin \theta \) is crucial in determining the contribution of each current element to the magnetic field.

### Step 3: Determine \( r \) and \( \theta \)
For a current element \( dl \) at position \( x \) along the segment \( PQ \):
- The distance \( r \) from the current element to the point \( x = +a \) is given by:
\[ r = a - x \]
- The angle \( \theta \) between \( dl \) (which is along the x-axis) and the line connecting the current element to the point \( x = +a \) is \( 0^\circ \) because both are in the same direction.

**Hint:** Check the geometry to confirm the angle and distance calculations.

### Step 4: Calculate \( \sin \theta \)
Since \( \theta = 0^\circ \):
\[ \sin \theta = \sin 0^\circ = 0 \]

**Hint:** Recall that the sine of zero is always zero, which will affect the calculation of the magnetic field.

### Step 5: Substitute into the Biot-Savart Law
Substituting \( \sin \theta = 0 \) into the Biot-Savart Law gives:
\[ dB = \frac{\mu_0}{4\pi} \frac{i \, dl \, \cdot 0}{(a - x)^2} = 0 \]

**Hint:** Recognize that if \( dB = 0 \) for each current element, the total magnetic field will also be zero.

### Step 6: Conclusion
Since the contribution to the magnetic field from each segment of the current-carrying wire is zero, the total magnetic field \( B \) at the point \( x = +a \) is:
\[ B = 0 \]

Thus, the magnetic field due to the section \( PQ \) at the point \( x = +a \) is \( 0 \).

**Final Answer:** The magnetic field at \( x = +a \) is \( B = 0 \).