# E&M—Lines of Force

was an almost completely self-educated physicist and chemist with a strongly intuitive mind. Apprenticed to a book binder at age fourteen, he had access to many scientific works. At the age of twenty, he attended a series of public lectures delivered by the eminent chemist Sir Humphry Davy, to whom he presented a three hundred page, bound book of his lecture notes. Subsequently Davy damaged his eyesight in a laboratory experiment, and hired young Faraday to be his secretary. After one of the Royal Institution's assistants was fired, at Davy's suggestion Faraday was appointed as Chemical Assistant on 1 March 1813.

Faraday's commoner origins were a source of personal consternation in dealing with other British scientists, who at the time were mostly gentlemen, Faraday was forced to eat with the servants at the Davy household. Once, when he and Davy toured Europe Faraday was forced to act as Davy's valet, which made Faraday miserable enough to contemplate retuning to England and giving up science altogether. Perhaps it was the memory of such mistreatment that later prompted him to reject knighthood and twice, the presidency of the Royal Society.

## Forces between charges and the Electric Field

By the time of Faraday, it was known that forces act between electric charges in a form similar to the way gravity acts: falling off as the inverse square of the distance between their centers. However, the electrical force could be attractive (between unlike charges) or repulsive (between like charges). Faraday saw a better way of representing the electric force. When a test charge (δq) experiences a force (δF) it is being affected by an otherwise invisible electric field (E).

The relation between an electric field and the force that it exerts on a test charge, δq, is given by the expression,

δF = δq E .                             (1a)

Operationally then, the electric field vector can be defined in terms of the force exerted on a test charge,

E = δF /δq.                             (1b)

Lines of force result naturally by using a positive test charge to detect an electric field. if you allow the electric force to pull the test charge along slowly, its trajectory will draw out a line of force. Note, that one must do that quasi-statically, i.e. without allowing the charge to accelerate and acquire a high enough speed to introduce inertial effects.

# Lines of Force

Intuitively, Faraday saw electrostatic force as being transmitted along continuous lines of force that branched only at points containing charges (Figs. 1a and b). Mathematically, the lines-of-force picture implies a field, which originates in positive and terminates on  negative charges. Incidentally, this characteristic of the lines-of-force picture also implies that charge is also a conserved quantity. The electric field vector at every location points along the tangent to the line of force in the direction of from positive to negative charge. The simplest pattern for the electric field is that of a single, isolated charge.

Fig 1a. Lines of Force about a Positive Charge point outward in radial directions.

fig. 1b. Lines of force about a Negative Charge point radially inward.

An isolated positive charge (Fig. 1a) is surrounded by a field consisting of electric lines of force, which at every point in space that is empty of charge, point outward radially in straight lines away from the location of the point charge, +Q. Fig 1a shows the lines of force depicting the field in a two dimensional cross section. In reality, the electric field points outward in three dimensions about the single point charge, +Q, the lines of force pass orthogonally through a set of concentric spherical surfaces.

The electric lines of force for a negative, isolated electric charge, have the same pattern as a positive charge, but they have a reversed direction, that is, they point inward toward the point charge, -Q (Fig. 1b).

At the time of Faraday's experiments, the discovery of the basic unit of charged had not been made, but the assumption could have been made that ordinary handfulls of electrically neutral matter contained vast quantities of positive and negative elementary, electrical charges in equal numbers. The universe considered in totality is electrically neutral. Each charge unit is balanced in the universe by its opposite charge. A minuscule imbalance in the number of elementary charges in a body would be enough to electrically charge it, the opposite charge having been conveyed to another body. On an electrically conducting sphere, an imbalanced charg would spread out in a uniform density over the surface of the sphere. You could think of the density of elementary charges as being proportional to the strength of an electric field emerging from this source, one line of force attached to each elementary charge. The strength of the electric field would be proportional to the number of lines of force emerging from the surface. Mathematically, we could write an equation for the electric field strength as follows:

E=  ur *constant*Q/(surface area of the sphere),      (2a)

or

E = ur *constant*Q/(4 π r2),                                      (2b)

or

E= ur * k*Q/ r2,,                                                          (2c)

where k is a constant into which the 4 π factor has been absorbed , and  ur is a unit vector (ur . ur = 1) pointing radially outward (ur = r/|r|).

Faraday pictured an electric field around an isolated positive charge (+Q) as radially straight lines of force (Fig. 1a) emerging uniformly in all directions about the charge. Any imaginary sphere in space otherwise empty of charges, having the charge +Q located as its center, would have the same number of lines of force crossing that surface, as if it were a conducting sphere on which the charge +Q is spread out uniformly on its surface.

The lines of force picture is consistent with the idea of an electrical force that acts between any two point charges with a strength proportional to the product of the charges and falling off with the inverse square of the distance between them. The lines of force represent an electric field (E) and the force (F) on a given charge (q) is simply,

F = q E                                                                   (3)

## Electric Field of a Charge Dipole

The electric field of an isolated, point, electric charge is very simply described by straight lines of force pointing outwardly from a positive charge (Fig. 1a)  and pointing inwardly about a negative charge (Fig. 1b). In this case, the lines of force have a spherical symmetry. Electric lines of force are not generally straight, but curved around a distribution of positive and negative point charges. The simplest configuration of this type is that of an electric dipole, which consists of two equal and opposite point charges separated by some distance along an axis, which is the dipole axis. In this case, the lines of force will have a cylindrical symmetry,  i. e., the field lines will look the same for any arbitrary rotation about the dipole axis.

Fig. 2. The electric field of two opposite and equal charges at any given point in space is the vector sum of the field from each charge.

Consider two equal and opposite charges: +Q (black) located at (-2.5,0,0) ;and -Q (red), at (2.5,0,0). The parentheses enclose the 3 spatial coordinates (x, y, z) for each point in question. The charges are separated 5 units along the x-direction (horizontal axis), which in this case is the dipole axis. Three electric field vectors are evaluated at the point (0, 2.5, 0): Epos (black, produced by the left, positive charge (+q, also black); Eneg (red), produced by the right charge (-q, also red); and the result of adding the former two vectors, which is the electric field vector for the dipole(Ed, green), evaluated at the point in question.

Fig. 3. Lines of force about a dipole of charges.

Starting at a small distance from the charge +q electric lines of force are mapped out by following the resultant electric vector for small distances, (Fig. 3). The resultant E vector is determined by the procedure described in Fig. 2, but using a two step process. If the small steps were taken along the originally computed E vector, the lines drawn out would systematically depart from the lines of force. For that reason, an initial computation and virtual step gives anew point to evaluate E again. The new E-vector averaged with the initial one gives a better approximation og the direction of the line of force (see ipython script).

Prof. Walter Lewin of MIT presents a lecture about the  Electric Field Lines in this video.

# Wrap up

Point electric charges, each of which would produce radially directed lines of force, in combination can produce curved lines of force, as we have seen in Fig. 3 for an electric dipole. In electricity and magnetism the more useful concept is not that of a force between electrically charged objects, but rather that of the electric and magnetic field, within which a charge (or magnet) experiences a force. In reality, even the minuscule imbalances of charges in matter that produce large-scale electric fields, involve a huge number of elementary charges. In another blog, we will show that the fields represent energy in space, which is available to charges or magnets introduced into that space.

## Python Code

At the command line enter the following code to set up the interactive python environment

ipython  --pylab

from pylab import *

# Once the ipython interpreter becomes active, then enter the following lines of code to define various functions

def E(q,x0,y0,z0,x,y,z):
if x ==0 and y==0 and z==0 or q==0: return [0.,0.,0.]
Ex = (x-x0)*q/sqrt((x-x0)*(x-x0)+(y-y0)*(y-y0)+(z-z0)*(z-z0))**3
Ey = (y-y0)*q/sqrt((x-x0)*(x-x0)+(y-y0)*(y-y0)+(z-z0)*(z-z0))**3
Ez = (z-z0)*q/sqrt((x-x0)*(x-x0)+(y-y0)*(y-y0)+(z-z0)*(z-z0))**3
return [Ex,Ey,Ez]

def circle(a, x0, y0, clr, flag):
delta=pi/100.
ang=arange(0., 2*pi+delta, delta)
x=x0+a*cos(ang)
y=y0+a*sin(ang)
plot(x,y,color=clr)
if flag ==1: p=fill(x,y,facecolor=clr)

def arrow(x, y, u, v, clr,lw, aang, eps):
dtor=pi/180.
x1=x+u
y1=y+v
plot([x,x1],[y,y1],color=clr, linewidth=lw)
rot=aang
up=eps*(u*cos(dtor*rot)-v*sin(dtor*rot))
vp=eps*(u*sin(dtor*rot)+v*cos(dtor*rot))
x2=x1+up
y2=y1+vp
plot([x1,x2],[y1,y2],color=clr)
rot=-aang
up=eps*(u*cos(dtor*rot)-v*sin(dtor*rot))
vp=eps*(u*sin(dtor*rot)+v*cos(dtor*rot))
x3=x1+up
y3=y1+vp
plot([x1,x3],[y1,y3],color=clr)
p=fill([x1,x2,x3,x1],[y1,y2,y3,y1], facecolor=clr)

def dipole(x0,y0,x,y,dx,dy,q):
x1=x0-0.5*dx
y1=y0-0.5*dy
x2=x0+0.5*dx
y2=y0+0.5*dy
r1=sqrt((x-x1)**2+(y-y1)**2)
r2=sqrt((x-x2)**2+(y-y2)**2)
Ex=q*((x-x1)/r1**3-(x-x2)/r2**3)
Ey=q*((y-y1)/r1**3-(y-y2)/r2**3)
print Ex, Ey
return [Ex,Ey]
#To generate Fig. 1a.delta=pi/4.
theta=arange(0., 2*pi, delta)

#------------------------------------------------------------------------------------------------------------

#Fig. 1a.

delta=pi/4.
theta=arange(0., 2*pi, delta)

q=1e5
a=10
dtor=pi/180.
figure()
p=fill([-10,10,10,-10,-10],[-10,-10,10,10,-10], facecolor='k')
for i in range(11):
circle(10.-i,0,0,'w',1)

for i in range(11):
circle(10.-i,0,0,'k',0)

for i in range(36):
a=10.
ang=10.*i   #  i*360./100.
x=a*cos(dtor*ang)
y=a*sin(dtor*ang)
arrow(0,0,x,y,'r',1, 170,0.1)
arrow(0.1*x, 0.1*y, 0.5*x, 0.5*y,'r',1, 170,0.25)

plot(0,0,'ko')

text(-0.05,0.1,'+Q', size=16, weight='bold', color='k')
title('Electric lines of force about a positive charge', size=20)
xlabel('X', size=18)
ylabel('Y', size=18)

#-------------------------------------------------------------------------------------------------------------

#Fig. 1b.

figure()
p=fill([-10,10,10,-10,-10],[-10,-10,10,10,-10], facecolor='k')
for i in range(11):
circle(10.-i,0,0,'w',1)

for i in range(11):
circle(10.-i,0,0,'k',0)

for i in range(36):
a=10.
ang=10.*i   #  i*360./100.
x=a*cos(dtor*ang)
y=a*sin(dtor*ang)
plot([0,x],[0,y],'r')
arrow(x,y,-0.5*x,-0.5*y,'r',1, 170,0.25)
arrow(x,y,-0.75*x,-0.75*y,'r',1, 170,0.1)

plot(0,0,'ko')
text(-0.05,0.1,'-Q', size=16, weight='bold', color='k')
title('Electric lines of force about a negative charge', size=20)
xlabel('X', size=18)
ylabel('Y', size=18)

#------------------------------------------------------------------------------------------------------------

# Fig. 2
q=1.0
x0=0
y0=0.0
z0=0.0
x=0.0
y=2.5
z=0
dx=5.0
dy=0.0
dz=0.0
figure()
#plot([-5,5],[0,0],'-b')          #Draw x-axis
#plot([0,0],[-5,5],'-',color='k') #Draw y-axis
plot([0,0],[-0.99, 3.49], 'y')
plot([-2.9,3.9], [0,0], 'y')
plot([-3,3.5], [-1,3.5], color='w')
plot(x0-dx/2.0,y0,'ko')                    #Mark position of left charge q.
text(x0-dx/2.-0.3,y0+0.5,'+q', size=24, weight='bold', color='k')
Ed=dipole(x0,y0,x,y,dx,dy,q)
Epos=E(q,x0-dx/2.,y0-dy/2.,z0-dz/2.,x,y,z)
v=Ed/sqrt(dot(Epos,Epos)) #Define unit vector along Ed.
arrow(x, y, v[0], v[1], 'g',1, 165, 0.2) #Draw arrow that points along Ed at (x,y,z)
v=Epos/sqrt(dot(Epos,Epos)) #Define unit vector along EL.
arrow(x, y, v[0], v[1], 'k',1, 165, 0.2) #Draw arrow that points along El at (x,y,z)

#Second charge
q=-1.0
x0=2.5
Eneg=E(q,x0,y0,z0,x,y,z)
v=Eneg/sqrt(dot(Epos,Epos)) #Define vector along EL scaled to unit vector for El.
plot(x0,y0,'ro')
text(x0,y0+0.5,'-q', size=24, weight='bold', color='r')
arrow(x, y, v[0], v[1], 'r',1, 165, 0.2) #Draw arrow that points along El at (x,y,z)
text(5,0.5,'X', size=24, weight='bold', color='k')
text(0.5,5,'Y', size=24, weight='bold', color='k')
text(-1.2,1.75,'Eneg', size=18, weight='bold', color='r')
text(-1.2,2.75,'Epos', size=18, weight='bold', color='k')
text(1.5,2.5,'Ed', size=18, weight='bold', color='g')
title('Electric field vectors from an electric dipole', size=20)
xlabel('X', size=18)
ylabel('Y', size=18)
circle(0.25,-2.5,0,'k',1)
plot(-2.5,0,'wo')
circle(0.25,2.5,0,'r',1)
plot(2.5,0,'wo')
#----------------------------------------------------------------------------------------------------------

#  Fig. 3

q=1.0
x0=0
y0=0.0
z0=0.0
x=0.0
y=2.5
z=0
dx=5.0
dy=0.0
dz=0.0
eps=0.1

figure()
#plot([-19,19],[0,0],'-y')          #Draw x-axis
#plot([0,0],[-40,40],'-y') #Draw y-axis
plot(x0-dx/2.0,y0,'ko')                    #Mark position of left charge q.
text(x0-dx/2.0-4.,y0+0.5,'+q', size=24, weight='bold', color='k')
plot(x0+dx/2.0,y0,'ro')
text(x0+3.,y0+0.5,'-q', size=24, weight='bold', color='r')
#text(5,0.5,'X', size=24, weight='bold', color='k')
#text(0.5,5,'Y', size=24, weight='bold', color='k')

xpos=x0-dx/2.
ypos=0.0
zpos=0.0
xneg=x0+dx/2.
yneg=0.0
zneg=0.0

for i in range(-15, 16):
ang=10.*i*pi/180.
x=xpos+0.5*cos(ang)
y=ypos+0.5*sin(ang)
z=0.0
dist=sqrt((x-xneg)**2+(y-yneg)**2+(z-zneg)**2)

while dist > 0.5 :

#Define dipold field vector along EL scaled to unit vector for El.
E=dipole(x0,y0,x,y,dx,dy,q)
v=E/sqrt(dot(E,E))
v=eps*v
if v[1] == 0.0: arrow(x, y, v[0], v[1], 'k',1., 150, 1)
E=dipole(x0,y0,x+eps*v[0],y+eps*v[1],dx,dy,q)
v1=E/sqrt(dot(E,E))
v[0]=0.5*(v[0]+v1[0])
v[1]=0.5*(v[1]+v1[1])
x1=x+eps*v[0]
y1=y+eps*v[1]
plot([x, x1], [y,y1], '-k')
x=x1
y=y1
dist=sqrt((x-xneg)**2+(y-yneg)**2+(z-zneg)**2)
x=-0.007
y=37.9592
v[1]=0
v[0]=1
arrow(x, y, v[0], v[1], 'k',1, 150, 1)
y=-y
arrow(x, y, v[0], v[1], 'k',1, 150, 1)
y=21.63
arrow(x, y, v[0], v[1], 'k',1, 150, 1)
arrow(x, -y, v[0], v[1], 'k',1, 150, 1)
y=14.2857
arrow(x, y, v[0], v[1], 'k',1, 150, 1)
arrow(x, -y, v[0], v[1], 'k',1, 150, 1)

plot([-14.5,14.5],[0.,0.], color='k')
plot(x0-dx/2.0,y0,'bo')                    #Mark position of left charge q.
plot(x0+dx/2.0,y0,'ro')
title('Lines of force about an electric dipole', size=24)
xlabel('X', size=18, weight='bold')
ylabel('y', size=18, weight='bold')

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