Download PHYS 3343 Lesson 1

PHYS 3323 Lesson 5
Electrostatics II
I.
Circulation of Electrostatic Fields
The electric field of a point charge does not circulate! This can be
easily seen if you look at the picture of the electric field lines of a
point charge.
q
We can also show this mathematically by taking the curl of a point
charge using the spherical representation of the curl. To make the
math easier to read, I will assume that the origin of my coordinate
system resides at the location of the point charge.

 1 q 
  E    
r̂ 
2 
 4πε0 r 

 E 
 

 rE   1   rE
E  1  1 E
E 
1  

θ
θ
r
  sin θ  E  
 r̂  
 
θ̂  
 r ˆ


r sin θ   θ 
  r  sin θ  
r  r  r
θ 


 E 

Since any electrostatic field is just the sum of the electric fields due to
individual point charges, it follows that all electrostatic fields have
zero circulation!

 E 0
This fact has some important consequences which we will now
examine.
II.
PHYS1224 Review
A.
Definition of Work

The work done on an object by a force F as the object is displaced
along some path from point A to point B is defined by the equation
 
W   F  dr
B
A
The work (value of the integral) may or may not depend on the path
that the object takes between point A and point B!!
If the work done by the force depends only on the initial and final
location of the object (i.e. its position) and not the path then the force
is said to be conservative.
If the work done by the force does depend on the path then the force
is said to be non-conservative.
You were told in PHYS1224 that a force was conservative if

  F 0 .
You should now realize from chapter 1 that this is a vector property
and applies to all vectors and not just forces!!
B.
Work-Energy Theorem
The work energy theorem is the central connection between our
energy concepts and Newton’s second law. It says that the work done
by the net external force (vector sum of all the external forces) acting
upon a body is equal to the change in the kinetic energy of the body.


WNet    Fi  d r  ΔK
B N
A i
C.
Potential Energy
We can now split our work terms into two sums depending upon
whether the forces are conservative or non-conservative.

 B

WNet   Fconservative  d r   Fnon-conservative  d r  ΔK
B
A
A
The first term is the net work by conservative forces while the second
term is the net work by non-conservative forces. Rearranging our
equation, we have
B



F

d
r

ΔK

F

d
r
non
conservati
ve
conservati
ve


B
A
A
The integral on the left hand side of the equation which is the work by
non-conservative forces can not be solved without specifying the
unique path traveled by the object.
The integral on the right hand of the equation depends only upon the
endpoints (the initial and final position of the object) and not upon the
path. Thus, someone else can choose a path and perform the integral
for you! Furthermore, we see by dimensional analysis that this
integral must be a change in energy. This leads us to define a new
type of energy – POTENTIAL ENERGY.
The negative of the work done by a conservative force is equal to the
negative of the change in the potential energy of a body due to that
force.
B

ΔU   Wconservative   Fconservative  d r
A
IMPORTANT POINTS
1.
Only changes in potential energy are defined.
It makes no sense to talk about potential energy at a point!! We
always mean the potential energy with respect to some arbitrary
zero potential energy reference point.
2.
III.
Potential energy functions can only be defined for
conservative forces!
Electrical Potential Energy and Electric Potential
We can now find the work done by an electric force on a test charge
as the test charge moves from point A to point B.
B
Q
A
B
 

W   F  d r  Q E  d r
B
A
A
We see that the work depends on the charge of our test object in the
same way that the electric force depends on its charge and the work
done on a test object by gravity depends on the test object’s mass.
We can remove this dependence using the same trick that we used to
obtain the electric field. In this case, we are finding a work per charge.
 
W
 E  dr
Q A
B

We can see that the work per charge done on our test charge for a
closed path is the circulation of the electric field!!
Wclosed path
Q



 E  dr
 EMF
Historically, this quantity is called the electromotive force (EMF)
which is a very unfortunate name since it is not a force or even a
vector!!
For electrostatic fields, the electric force is conservative (i.e. electric
fields do not circulate). In this case, it is meaningful to talk about the
change in the electrical potential energy of our test charge. From our
work above, we see the change in the electrical potential energy per
charge of the test charge as it moves from point A to point B is given
by
B

ΔU - W

   E  dr
Q
Q
A
This ratio of the change in the electric potential energy divided by the
test charge is called the change in the electric potential or voltage!
ΔV 
ΔU
Q
You should note that all of the properties mentioned previously about
potential energy functions effect electrical potential functions since
electric potential is defined in terms of electrical potential energy!
We also decided to name a new unit. The units of electrical potential
by dimensional analysis are Joules/Coulombs. A Joule/Coulomb is
called a volt after Alexandra Volta who developed the battery.
IV.
Connection Between Electrical Potential and Electric Fields
From our work in the previous section, we see that one can obtain the
electrical potential difference between two points in space if you
know the electric field using the formula
 
ΔV    E  d r
B
A
This formula is useful for developing relationships for the electrical
potential of simple devices where the electric field can be calculated
by Gauss’ law.
However, the formula is not used when making real world
calculations! This is because the reason for using electric potential is
to avoid the mathematical difficulty in solving the vector integral
required to find the electric field! Electrical potential is a scalar and so
the math including integration is often simpler.
If we know or can calculate the change in electrical potential, how do
we find the electric field? The answer is provided in chapter 1. The
change in a function between two points is equal to the integral of the
gradient of that function along any path between the two points!

ΔV   V  d r
B
A
By comparison with our previous formula, we then have the following
relationship between the electric potential and an electrostatic field

E   V
It is important to note that this formula is only true for a conservative
electric field (i.e. electrostatics!). It is probably a good time to reread
both your PHYS2424 textbook’s chapter on electric potential and
chapter of your E&M text so that you see all the connections from
different perspectives.
V.
Electric Potential of a Test Charge
You should memorize the formula for the electric potential due to a
single point charge. This is because more complicated systems can be
considered to be the sum of the contributions due to a collection of
point charges.
Vr  
1 q
4πε0 r
Proof: Consider a point charge q located at the origin and assume that
our zero electrical potential reference point is at r = .
r=
r
q
VI.
Electrical Potential for a Collection of Point Charges
A.
Discrete Point Charges
The total electric potential due to a collection of point charges is given
by
Vr  
1
4πε 0
N
q
i

r
i 1 i
This is a scalar equation and easier to calculate than for electric fields.
B.
Continuous Charge Distribution
The total electric potential for a continuous charge distribution is
given by
Vr  
1
4πε 0
dq
 r
all charges
If the charge is spread throughout a volume, we would write the
infinitesimal differential charge dq in terms of the charge density and
our formula becomes
Vr  
1
4πε 0

all space
ρr'dv'
r
We can integrate over all space and not just the charge volume since
the additional volumes where the charge density is zero don’t add to
the integral.
VII. Poisson’s Equation and Laplace’s Equation
We will run into many problems in which we will not know charge
density or where we will want to avoid solving integrals. For instance,
if a charge is near a conductor it will induce the charges on the
conductor to redistribute in a way that we don’t know a priori.
Thus, it would be useful to find a differential equation for the electric
potential that we could solve instead of an integral equation.
We start by considering the divergence of the electric field (Gauss’
law)
 ρ
E
ε0
For electrostatic fields, we can insert our relationship between the
electric field and the electric potential
   V
ρ
ε0
Thus, we have Poisson’s equation.
2V  
ρ
ε0
If there is no charge density in the region where you are solving for
the electric potential, Poisson’s equation reduces to the simple
Laplace’s equation.
2V  0
We will study some special techniques for solving these equations in
chapter 3.