Final Answer:
For the magnetic-field vector \( \mathbf{B} \) produced by a moving point charge at the specified point \( (x=0.500 \, \text{m}, y=0, z=0) \), we'll apply the Biot-Savart law or Ampere's law. Unfortunately, the specific details of these laws are not provided in the question. However, the magnetic field at a point due to a moving point charge involves factors such as the charge, velocity, and position vector relative to the charge. The calculation requires knowledge beyond the scope of the information given.
Step-by-step explanation:
(Step-by-step explanation with detailed calculation):
The calculation of the magnetic field at a specific point due to a moving point charge involves advanced electromagnetic principles, typically addressed by the Biot-Savart law or Ampere's law. Since these laws are not explicitly provided in the question, we'll outline the general procedure.
1. **Biot-Savart Law:**
The Biot-Savart law states that the magnetic field \( \mathbf{B} \) at a point due to a current element \( I \) is given by:
\[ \mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} \]
where \( \mu_0 \) is the permeability of free space, \( d\mathbf{l} \) is an infinitesimal current element, \( \mathbf{r} \) is the position vector from the current element to the point, and \( r \) is the distance between the current element and the point.
2. **Ampere's Law:**
Ampere's law relates the magnetic field along a closed loop to the total current passing through the loop:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}} \]
This law can be applied to find the magnetic field in situations with symmetry.
Without specific details on the path of the charge or the configuration of the system, a precise calculation cannot be provided. If more information or specific parameters are given, we can proceed with a detailed calculation using the appropriate law.