Biot-Savart law: magnetic field of a straight wire

Magnetism and magnetism related phenomena are fascinating almost for everyone, in fact, I remember being a intrigued by the interaction between magnets since I was a little kid. On the other side, Python is great for plotting vector fields, although now that I am using also Matlab I found out most functions in numpy and matlab are similar to those used in Matlab (I don’t know who got inspired by who though).

Here is a small sketch of a vector field. Remember the Biot-Savart law for a straight infinitely long wire, which (assuming the current is flowing upwards) looks something like this:

Below is a 3d representation of the vector field and the Python code used to generate it. The tricky part (if I’m allowed to say so) is that you need to convert the value of the vector field from cylindrical coordinates into cartesian coordinates in order to plot it. Perhaps there’s a way to plot a vector field using also spherical and cylindrical coordinates however I still do not know how to do that.

	from mpl_toolkits.mplot3d import axes3d
	import matplotlib.pyplot as plt
	import numpy as np
	x = np.linspace(-4,4,10)
	y = np.linspace(-4,4,10)
	z = np.linspace(-4,4,10)
	x,y,z = np.meshgrid(x,y,z)
	# 3d figure
	fig = plt.figure()
	ax = fig.gca(projection='3d')
	def B(x,y):
	i = 1 #Amps in the wire
	mu = 1.26 * 10**(-6) #Magnetic constant
	mag = (mu/(2*np.pi))*(i/np.sqrt((x)**2+(y)**2)) #Magnitude of the vector B
	by = mag * (np.cos(np.arctan2(y,x))) #By
	bx = mag * (-np.sin(np.arctan2(y,x))) #Bx
	bz = z*0 #Bz (zero, using the right-hand rule)
	return bx,by,bz
	def cylinder(r):
	phi = np.linspace(-2*np.pi,2*np.pi,100)
	x = r*np.cos(phi)
	y = r*np.sin(phi)
	return x,y
	# Plot of the fields
	bx,by,bz = B(x,y) #Magnetic field
	cx,cy = cylinder(0.2) #Wire
	# Plot of the 3d vector field
	ax.quiver(x,y,z,bx,by,bz,color='b',length=1,normalize=True)
	#Plot the magnetic field
	for i in np.linspace(-4,4,800): #Plot the wire
	ax.plot(cx,cy,i,label='Cylinder',color='r')
	plt.title('Magnetic field of a straight wire')
	plt.xlabel('x')
	plt.ylabel('y')
	plt.show()

view raw magnetic_field.py hosted with ❤ by GitHub

Hope this was interesting! :)