This notebook shows how to numerically calculate and visualise the fields around a circular homogeneously charged insulating plate. (C) Jo Verbeeck, EMAT, University of Antwerp, sept 2019
load the required libraries for calculation and plotting
In [2]:
import numpy as np
import matplotlib.pyplot as plt
Create a polar grid to describe infinitesimal charged areas that make up the plate
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rpoints=100 #nr of radial points on the plate
thetapoints=100 #number of azimuthal steps
rmax=1 #extension of grid [m]
pref=9e9 # 1/(4pi eps0)
sigma=1 #surface charge density [C/m^2]
r=np.linspace(rmax/rpoints,rmax,rpoints)
dr=r[2]-r[1]
theta=np.linspace(2*np.pi/thetapoints,2*np.pi,thetapoints) #careful, avoid double counting of theta=0 and theta=2pi
dtheta=theta[2]-theta[1]
[r2d,theta2d]=np.meshgrid(r,theta,indexing='ij') #2D matrices holding x or y coordinate for each point on the grid
#cartesian coordinates for each element in the plate
x2dp=np.multiply(r2d,np.cos(theta2d))
y2dp=np.multiply(r2d,np.sin(theta2d))
Define a grid for the plane in which we want to get the field information (is 3D but this is hard to visualise so we choose a single 2D plane perpendicular to the disc and cutting through the center of the disc
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xpoints=100
zpoints=100
xmax=2*rmax
zmax=xmax
x=np.linspace(-xmax,xmax,xpoints)
z=np.linspace(-zmax,zmax,zpoints)
x2d,z2d=np.meshgrid(x,z)
Now perform the integration over the disc summing up all the field stemming from the infinitesimal charged areas using the superposition principle. For clarity I will use the naive way using for loops. This is ridicoulously slow in an interpreted language as python, but it is easier to understand what is happening.
In [55]:
ex=np.zeros(x2d.shape)
ez=np.zeros(x2d.shape)
for rid in range(rpoints):
for thetaid in range(thetapoints):
dq=sigma*r[rid]*dr*dtheta #infinitesimal charge in this segment
rx=x2dp[rid,thetaid]-x2d #keep track of the vector connecting the segment with any point on the xz grid
ry=y2dp[rid,thetaid]-0
rz=0-z2d
dist=np.sqrt(np.square(rx)+np.square(ry)+np.square(rz)) #distance of this segment to any point in the xz grid
ex=ex+pref*dq*np.multiply(np.power(dist,-3),rx)
ez=ez+pref*dq*np.multiply(np.power(dist,-3),rz)
#ey #zero because of symmetry (the plane cuts the disc in half)
e=np.sqrt(np.square(ex)+np.square(ez))
And now its showtime!
In [56]:
plt.imshow(e,extent=[-xmax, xmax, -xmax, xmax])
plt.title('electric field and fieldlines')
plt.xlabel('x');
plt.ylabel('z');
plt.streamplot(x2d,z2d,ex,ez)
plt.axis('square')
plt.colorbar
plt.show()
Note how far away from the plate, all fieldlines behave as if the plate was a point charge. Only closer to the plate, the field differs. Try to make rmax>xmax (e.g. xmax=0.5*rmax) larger and see how the field starts to become independent from the distance to the plate (fieldlines become more parallel), but only if this distance stays much smaller than de radius of the plate. Such a homogeneous field would store an infinite amount of energy but it would also hold an infinite amount of charge, but as infinite plates don't occur in reality, this is no problem. Note how the field lines don't arrive perpendicular to the plate surface. This is expected for an insulator as assumed here, but would be different if the plate would be a conductor (in that case the field lines would arrive perpendicular, but the charge density would not be homogeneous as the charges would prefer to sit on the outside rim of the plate).