Scalar Potential:

\\[ \phi(\mathbf{r}, t) = \frac{1}{4\pi \epsilon_0} \frac{q}{|\mathbf{r} – \mathbf{r}_s(t_r)| (1 – \mathbf{\hat{n}} \cdot \boldsymbol{\beta}(t_r))} \\]

Vector Potential:

\\[ \mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \frac{q \mathbf{v}(t_r)}{|\mathbf{r} – \mathbf{r}_s(t_r)| (1 – \mathbf{\hat{n}} \cdot \boldsymbol{\beta}(t_r))} \\]

Retarded Time Condition:

\\[ t_r = t – \frac{|\mathbf{r} – \mathbf{r}_s(t_r)|}{c} \\]

Definitions:

\\[ \mathbf{\hat{n}} = \frac{\mathbf{r} – \mathbf{r}_s(t_r)}{|\mathbf{r} – \mathbf{r}_s(t_r)|}, \quad \boldsymbol{\beta} = \frac{\mathbf{v}}{c} \\]

Where:

• \\( \phi \\): Scalar potential (V)
• \\( \mathbf{A} \\): Vector potential (T·m/A)
• \\( q \\): Charge (C)
• \\( \mathbf{v} \\): Velocity of the charge (m/s)
• \\( \mathbf{r} \\): Observation point (m)
• \\( \mathbf{r}_s(t_r) \\): Source position at retarded time (m)
• \\( t_r \\): Retarded time (s)
• \\( c \approx 2.99792458 \times 10^8 \, \text{m/s} \\): Speed of light
• \\( \epsilon_0 = 8.854187817 \times 10^{-12} \, \text{F/m} \\): Permittivity of free space
• \\( \mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A} \\): Permeability of free space

Solve for retarded time: \\( t_r = 0 – \frac{\sqrt{(v_x t_r)^2 + 0.1^2}}{c} \\)

\\[ t_r \approx -3.3356 \times 10^{-10} \, \text{s} \\] \\[ R = \sqrt{(1 \times 10^7 \cdot (-3.3356 \times 10^{-10}))^2 + 0.1^2} \approx 0.1 \, \text{m} \\] \\[ \mathbf{\hat{n}} \cdot \boldsymbol{\beta} = \frac{-1 \times 10^7 \cdot (-3.3356 \times 10^{-10})}{0.1} \cdot \frac{1 \times 10^7}{2.99792458 \times 10^8} \approx 0.03336 \\] \\[ \phi = \frac{1}{4\pi \cdot 8.854187817 \times 10^{-12}} \cdot \frac{1 \times 10^{-9}}{0.1 \cdot (1 – 0.03336)} \approx 92.52 \, \text{V} \\] \\[ A_x = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{1 \times 10^{-9} \cdot 1 \times 10^7}{0.1 \cdot (1 – 0.03336)} \approx 1.034 \times 10^{-4} \, \text{T·m/A} \\]