Multipole Expansion Calculator
Multipole Expansion Calculator computes potential V for a charge, visualizing vs. distance.
Formulas Used in Multipole Expansion Calculator
The calculator computes the electrostatic potential for a point charge using the multipole expansion (monopole and dipole terms):
Electrostatic Potential:
\\[ V(\mathbf{r}) \approx \frac{1}{4\pi \epsilon_0} \left[ \frac{Q}{r} + \frac{\mathbf{p} \cdot \hat{r}}{r^2} \right] \\]Where:
- \\( V \\): Electrostatic potential (V)
- \\( Q = q \\): Monopole moment (total charge, C)
- \\( \mathbf{p} = q \mathbf{r}_q \\): Dipole moment (C·m), with \\(\mathbf{r}_q = (x_q, y_q, z_q)\\)
- \\( r \\): Distance from origin along x-axis (m)
- \\( \hat{r} \\): Unit vector (here, \\(\hat{x}\\))
- \\( \epsilon_0 = 8.854187817 \times 10^{-12} \, \text{F/m} \\): Permittivity of free space
For a single charge at \\(\mathbf{r}_q\\), along the x-axis (\\(\mathbf{r} = r \hat{x}\\)):
\\[ V(r) = \frac{1}{4\pi \epsilon_0} \left[ \frac{q}{r} + \frac{q x_q}{r^2} \right] \\]Example Calculations
Example 1: 1 nC at (0.01, 0, 0), r = 0.1 m
Input: \\( q = 10^{-9} \, \text{C}, x_q = 0.01 \, \text{m}, y_q = 0, z_q = 0, r = 0.1 \, \text{m} \\)
\\[
V = \frac{1}{4\pi \cdot 8.854187817 \times 10^{-12}} \left[ \frac{10^{-9}}{0.1} + \frac{10^{-9} \cdot 0.01}{0.1^2} \right] \approx 98.864 \, \text{V}
\\]
Result: \\( V \approx 98.864 \, \text{V} \\)
Example 2: 1 nC at (0.01, 0, 0), r = 0.5 m
Input: \\( q = 10^{-9} \, \text{C}, x_q = 0.01 \, \text{m}, y_q = 0, z_q = 0, r = 0.5 \, \text{m} \\)
\\[
V = \frac{1}{4\pi \cdot 8.854187817 \times 10^{-12}} \left[ \frac{10^{-9}}{0.5} + \frac{10^{-9} \cdot 0.01}{0.5^2} \right] \approx 18.155 \, \text{V}
\\]
Result: \\( V \approx 18.155 \, \text{V} \\)