Step 1: Concept Identification: This problem necessitates the application of the Biot-Savart Law to determine the magnetic field generated by a minute current-carrying segment. The Biot-Savart Law quantifies the magnetic field at a specific spatial point resulting from a current element.
Step 2: Governing Equation: The Biot-Savart Law defines the magnitude of the magnetic field \(dB\) produced by a current element \(Id\vec{l}\) at a distance \(r\) as follows: \[ dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2} \] Here, \(\mu_0\) represents the permeability of free space (\(4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\)), \(I\) is the current, \(dl\) is the length of the current element, \(r\) is the distance from the element to the observation point, and \(\theta\) is the angle between the current element's direction and the position vector to the observation point.
Step 3: Detailed Calculation: Input Parameters: Current, \(I = 10 \, \text{A}\). Element length, \(dl = 1 \, \text{cm} = 0.01 \, \text{m}\). Distance to the point on the y-axis, \(r = 0.5 \, \text{m}\). The current element is oriented along the x-axis, and the observation point lies on the y-axis. Consequently, the angle \(\theta\) between the current element \(d\vec{l}\) and the position vector \(\vec{r}\) is \(90^\circ\). Thus, \(\sin\theta = \sin(90^\circ) = 1\). The constant \(\frac{\mu_0}{4\pi} = 10^{-7} \, \text{T}\cdot\text{m/A}\). Computation: Substituting the given values into the Biot-Savart Law formula: \[ dB = (10^{-7}) \frac{(10 \, \text{A}) \times (0.01 \, \text{m}) \times 1}{(0.5 \, \text{m})^2} \] \[ dB = 10^{-7} \frac{0.1}{0.25} \] \[ dB = 10^{-7} \times 0.4 \] \[ dB = 4 \times 10^{-8} \, \text{T} \]
Step 4: Conclusive Result: The magnitude of the magnetic field produced by the current element at the specified point is \(4 \times 10^{-8}\) T.