In [1]:
import numpy as np
from matplotlib import pyplot as plt
%matplotlib inline
Magnetic field produced by a dipole is (see wikipedia):
$$\mathbf{B}({\mathbf{r}}) = \frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{r^{5}} - \frac{{\mathbf{m}}}{r^{3}}\right)$$According to WolframAlpha: $$\mu_{0} = \frac{\pi}{2.5 \cdot 10^6}\ \frac{\mathrm{N}}{\mathrm{A}^2}$$
$$\frac{\mu_{0}}{4\pi} = \frac{\pi}{4 \pi \cdot 2.5 \cdot 10^6} = 10^{-7}\, \frac{\mathrm{H}}{\mathrm{m}}$$In [4]:
def dipole(m, r, r0):
"""Calculate a field in point r created by a dipole moment m located in r0.
Spatial components are the outermost axis of r and returned B.
"""
# we use np.subtract to allow r and r0 to be a python lists, not only np.array
R = np.subtract(np.transpose(r), r0).T
# assume that the spatial components of r are the outermost axis
norm_R = np.sqrt(np.einsum("i...,i...", R, R))
# calculate the dot product only for the outermost axis,
# that is the spatial components
m_dot_R = np.tensordot(m, R, axes=1)
# tensordot with axes=0 does a general outer product - we want no sum
B = 3 * m_dot_R * R / norm_R**5 - np.tensordot(m, 1 / norm_R**3, axes=0)
# include the physical constant
B *= 1e-7
return B
In [5]:
X = np.linspace(-1, 1)
Y = np.linspace(-1, 1)
Bx, By = dipole(m=[0, 0.1], r=np.meshgrid(X, Y), r0=[-0.2,0.1])
plt.figure(figsize=(8, 8))
plt.streamplot(X, Y, Bx, By)
plt.margins(0, 0)
Works in 3 dimensions as well:
In [7]:
dipole(m=[1, 2, 3], r=[1, 1, 2], r0=[0, 0, 0])
Out[7]: