Lienard-Wiechert Potentials Calculator

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

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