1. Setup Definition:
- Conductor length: 1 cm (0.01 m), centered at origin along x-axis, from \( -0.005\,\text{m} \) to \( +0.005\,\text{m} \).
- Current: \( I = 1\,\text{A} \)
- Observation point: \( (1, 1, 0) \)
2. Magnetic Field Calculation (Finite Segment):
Employing the Biot–Savart law for a finite straight conductor:
\[ \vec{B} = \frac{\mu_0 I}{4\pi r} (\sin\theta_1 + \sin\theta_2)\hat{n} \]
Parameters:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{Tm/A} \) (permeability of free space)
- \( r \) = perpendicular distance from wire to point = \( \sqrt{(1)^2 + (1)^2} = \sqrt{2} \, \text{m} \)
- Due to symmetry, \( \theta_1 = \theta_2 \), angles measured from the wire's center to its ends.
- \( \tan\theta = \frac{L/2}{r} = \frac{0.005}{\sqrt{2}} \Rightarrow \theta \approx \tan^{-1}(0.0035) \approx 0.2^\circ \) (negligibly small)
- For small angles, \( \sin\theta \approx \theta \text{ (in radians)} \), thus \( \sin\theta_1 + \sin\theta_2 \approx 2\theta \)
Approximate magnetic field magnitude:
\[ B = \frac{4\pi \times 10^{-7} \times 1}{4\pi \times \sqrt{2}} \times 2 \times \frac{0.005}{\sqrt{2}} = \frac{10^{-7} \cdot 2 \cdot 0.005}{2} = \frac{10^{-7} \cdot 0.005}{\sqrt{2}} = \frac{5.0 \times 10^{-10}}{\sqrt{2}} \, \text{T} \]
3. Conclusion:
Option (C) \( \frac{5.0 \times 10^{-10}}{\sqrt{2}} \, \text{T} \) is the correct magnetic field value.