Visualisation of a generic multipole

This notebook shows how to numerically calculate and visualise the fields around an electrostatic multipole. Questions:

do you see local minima or maxima in the potential? (would a 3D generalisation be better?)
what happens if npoles get higher and higher? inside and outside?

load the required libraries for calculation and plotting



In [1]:

    
import numpy as np
import matplotlib.pyplot as plt

create grid to plot (choose 2D plane for visualisation cutting through charge centers , but calculation is correct for 3D)



In [13]:

    
xpoints=512 #nr of grid points in 1 direction
xmax=1 #extension of grid [m]
pref=9e9 # 1/(4pi eps0)
x=np.linspace(-xmax,xmax,xpoints)
y=x
[x2d,y2d]=np.meshgrid(x,y,indexing='ij') #2D matrices holding x or y coordinate for each point on the grid

#define multipole
npoles=6 #number of poles, needs to be even
fradius=0.5*xmax #field radius
sradius=0.1*xmax #radius of spheres making up the multipole
vamp=1 #voltage amplitude on multipole (half of the poles have +vamp, other half has -vamp)

calculate the potential of the set of spheres (use a function that we can reuse later)



In [14]:

    
def multipolepotential(x,y,z,npoles,v,fradius,sradius):
    #assume a set of n conducting spheres of radius on a circle of radius fradius (field radius)
    #npoles is number of poles and needs to be even >0
    #the spheres are positioned in the xy plane and have a potential of V for the even spheres and -V for the odd spheres
    out=np.zeros(x.shape)
    potentialin=np.zeros(x.shape)
    potential=np.zeros(x.shape)
    theta=np.linspace(0,2*np.pi,npoles+1)
    if(npoles % 2) == 0:
        for nid in range(npoles):
            #make a superposition of the potential for each of the spheres
            vin=v*(-1.0)**nid            
            xn=fradius*np.cos(theta[nid])
            yn=fradius*np.sin(theta[nid])
            r=np.sqrt(np.square(x-xn)+np.square(y-yn)+np.square(z)) #distance to sphere n
            in1=r<sradius  #logical function 1 if inside sphere, 0 if outside
            out1=r>=sradius #logical function 0 if inside sphere, 1 if outside
            potential=potential+vin*sradius*np.multiply(np.power(r,-1),out1) 
            out=out+out1
            potentialin=potentialin+vin*in1  
        #do a rescaling to match potential as the superposition changes the actual potential on the spheres slighlty
        idin=np.where(potentialin)
        idout=np.where(out)
        potential[idin]=potentialin[idin]
        potential[idout]=v*(potential[idout]/np.max(potential[idout]))
    else:
        potential=None
        #undefined
    return potential

v=multipolepotential(x2d,y2d,np.zeros(x2d.shape),npoles,vamp,fradius,sradius)
ex,ey=np.gradient(-v,x,y)  #strange ordering due to meshgrid
e=np.sqrt(ex**2+ey**2)

And now its showtime!



In [15]:

    
#show vector plot, but limit number of points to keep the number of vector reasonable
skippts=20
skip=(slice(None,None,skippts),slice(None,None,skippts)) #dont plot all points in a quiver as this becomes unreadable
plt.quiver(x2d[skip],y2d[skip],ex[skip],ey[skip])
plt.title('electric field')
plt.xlabel('x')
plt.ylabel('y')
plt.axis('square')









    Out[15]:





(-1.0978473581213306,
 1.054794520547945,
 -1.0978473581213306,
 1.054794520547945)

Note how the field emanates from the positive charge sinks into the negative charge



In [17]:

    
plt.imshow(e,extent=[-xmax, xmax, -xmax, xmax])
plt.title('electric field and fieldlines')
plt.xlabel('x');
plt.ylabel('y');
plt.streamplot(x2d,x2d,ey,ex)
plt.axis('square')
plt.colorbar
plt.show()

Note the interesting npoles/2 fold symmetry of the field



In [11]:

    
plt.imshow(v,extent=[-xmax, xmax, -xmax, xmax]) 
plt.title('electrostatic potential V')
plt.xlabel('x')
plt.ylabel('y')
plt.axis('square')
plt.colorbar()
plt.show()



In [12]:

    
nlines=50;
plt.contour(x2d,y2d,v,nlines)
plt.title('equipotential surfaces')
plt.xlabel('x')
plt.ylabel('y')
plt.axis('square')
plt.colorbar
plt.show()

Equipotential lines are always perpendicular to the fieldlines.