Lecture #3-4
Electric Potential
Today we are going to look at electrostatic problems from a different stand point.
We will use the same idea which we have developed in classical mechanics. As you may
recall, we first studied physical behavior of mechanical systems using vector quantities
such as force, acceleration and velocity. This was rather complicated method due to the
vector nature of these quantities. We then moved on and considered energy
transformations in the systems. Since energy is a scalar which is not that sensitive to the
choice of coordinate frame, it turns out that many (almost unsolvable by means of
Newton’s laws) problems can be resolved easier using energy consideration. Our task
today will be to see how similar ideas, related to the concept of energy, are working in
electrostatics.
We have learned that, in addition to mechanical forces, there is also electrostatic
force acting between any two charged particle-like objects. It is very similar to
gravitational force. In both cases these are long range forces. Also, according to the
Coulomb’s law, electrostatic force acting between the two charged particle-like objects is
directed along the line connecting these objects and proportional to the absolute value of
each of the interacting charges and inversely proportional to the square of the distance
between the charges. Since it is so similar to the Newton’s law of gravitation, we can
guess that it should also be a similarity between the potential energy of gravitational
interaction and electrostatic interaction.
We have also introduced the concept of electric field which again is very similar
to the concept of gravitational field. According to our definition, the electric field
E
F
q
(3.4.1)
is the electrostatic force acting per unit of positive testing charge. It is a vector and it is
measured in N/C. It is the same as calculating gravitational force produced on the unit
mass. We know that gravitational field near the earth’s surface is almost constant and

equal g with the absolute value of 9.8 m s 2 . We used this fact in order to introduce the
potential energy of gravitational field.
Let us try to recover the general ideas about potential energy and see how they will
work in application to electric field. Considering some object (for instance a charged
particle), even if this object is at rest and has zero kinetic energy, we may say that it is
still potentially capable of performing work. For instance, if we place this charged
particle inside of the uniform electric field of the parallel plate capacitor, there will be
electrostatic force acting on it from the field. So, if we release this charge it will start
moving. In this situation the external electric field will perform work on the charge
transferring potential energy of electric field into kinetic energy of the moving charge.
Here, as well as in classical mechanics, we are dealing with the concept of potential
energy PE which depends on configuration (or arrangement) of the system. In our
example it depends on original position of the charge relative to the plates of the
capacitor. For instance, if we had a positive charge and placed it close to the positively
charged plate, then it will be pushed away from this plate and, at the same time, be
attracted by the negative plate, so it will be moving towards the negatively charged plate.
On the other hand, if we put it on the surface of the negatively charged plate at the
beginning of the experiment it will not move at all. The picture will be exactly opposite if
we do this with a negative charge. So, in the case of the electric field, potential energy
depends on both configuration of the system as well as the sign of the charge.
If we move a positive charge in the direction opposite to the direction of the electric
field then electric field does negative work, which is
W   Fd  qEd
(3.4.2a)
If a positive charge moves in the direction of the electric field then this field does positive
work of
W  Fd  qEd
(3.4.2b)
The negative work in the first case transfers energy from the kinetic energy of the charge
into potential energy of the charge-capacitor system. In the second case, when the
positive charge moves down to the negatively charged plate, its kinetic energy grows. It
comes from the potential energy of the charge-capacitor system. So, we can define the
change of electric potential energy, PE , of this system as the negative of work, W,
done by the electrostatic force
PE  W .
(3.4.3)
This equation 3.4.3 is the same as for any other potential field.
In all the situations
1) The system consists of two or more objects.
We always have at least two objects (a box and the earth or a box and the spring or a
point-like charge and the capacitor). However, it can be more than two objects in the
system. (For instance, we can consider a system consisting of several point-like charges.)
2) The force acts between the particle-like object (the box or the charge) and the
rest of the system.
So we can say that this object is under the influence of the potential field from the rest of
this system.
3) When configuration of the system changes, the force does work on the particlelike object, transferring energy between the kinetic energy of the object and the
potential energy of the system.
4) When configuration of the system is reversed, the force reverses the energy
transfer by performing the same in absolute value but opposite in sign work,
bringing system back to its original energetic state.
If the work of the force when we first changed configuration of the system is W1 , and
when the system is brought back to its original state is W2 , then our last statement means
that W1  W2 . Forces, for which this relation is true, are called conservative forces.
Electrostatic force is a conservative force.
As we know, the existence of conservative forces provides significant
simplification to solutions of some problems. If we consider a charged particle moving in
the electric field along the closed path beginning at some initial position and then coming
back to the same position, the shape of this path (particle's trajectory) does not matter. Or
in other words: The net work done by a conservative electrostatic force on a particle
moving around every closed path is zero. The important consequence of this statement is
that the work done by electrostatic force on a particle moving between the two points
does not depend on the shape of the path taken by the particle. So, if the particle moves
in a field of conservative force from point a to point b, whatever path it takes 1 or 2, the
work done by the conservative force on this particle on both paths is the same as long as
points a and b are the same
Wab,1  Wab,2 .
(3.4.4)
This allows solving problems, which involve very complicated paths by calculating work
along much simpler path, but between the same original and final points.
In a same way, as we did when we introduced electric field by means of equation
3.4.1, we can consider change of electric potential energy per unit of charge, which is
called electric potential difference (or voltage when used in application to electric
currents)
V 
PE
W

q
q
(3.4.5)
Even though only change of potential energy and potential difference has physical
significance, one can still introduce absolute values for those quantities by picking the
appropriate reference level of energy or potential. In this situation one can define the
electric potential at given point in space as taken with ”minus” sign work done by
electric field when a positive unit charge is moved to this point from the ground level of
the potential.
As any other physical quantity, electric potential has its own units. The SI unit of
electric potential is called Volt which is 1V 
1J
.
1C
Note that as it was in the case of the electric field, where the field itself does not
depend on the testing charge, the electric potential depends on configuration of the
system but it does not depend on the absolute value of the testing charge.
If we continue with the example of the uniform electric field inside of the plane
capacitor, then we discover a simple relationship between the electric potential and the
electric field. Indeed, if we have a testing charge q moving in the direction of electric
field along axis x perpendicular to the pates of the capacitor then
W
qEx

  Ex,
q
q
V
E
x
V  
(3.4.6)
This means that electric field is the rate of change of electric potential. Even though we
just have proved this fact for uniform electric field inside of the capacitor, it has very
general nature and can be applied to almost any situation with the correction that the
electric field and the rate of change of electric potential are not necessarily constants.
When we discussed electric field, we have introduced a useful concept of the
electric field lines. These lines are tangential to the direction of the electric field vectors
at any given point in space. Using similar ideas we can introduce equipotential lines (in 2
dimensions) or equipotential surfaces (in 3 dimensions). These lines (or surfaces) pass
through all the points in space where electric potential has the same value. It is easy to
see that electric field lines are perpendicular to equipotential surfaces. The distances
between the equipotential surfaces become smaller as electric field gets higher. Inside of
the parallel plate capacitor, where the field is uniform, all equipotential surfaces are
parallel to plates and the distances between these equipotential surfaces are the same for
the same potential change.
The fact that electric field is a conservative field allows us to solve many
problems using conservation of energy. A good example is the motion of electric charge
inside of the parallel plate capacitor or any other uniform electric field.
According to the Coulomb’s law the electrostatic field produced by a point-like
charge q1 has the magnitude of
E
kq1
r2
(3.4.7)
This is the same as behavior of the gravitational field of the point-like or spherical object.
As we remember, the electric field outside of the spherical charged object is the same as
the electric field produced by a point-like object as if all the charge was concentrated at
the center of the sphere. This is the content of the Newton’s shell theorem. The same will
be true for electric potential. If we set the ground level of potential at infinity then
V
1 q1
,
4 0 r
(3.4.8)
Principle of superposition which we have used for forces and electric fields can be
generalized to the case of the potentials as well. However, in contrast to electric force or
electric field, the electric potential is a scalar. So, if we want to calculate the total electric
potential produced by a system of the point-like charges, we have to add algebraically all
electric potentials produced by different point-like charges.
Combining equation 3.4.8 with the principle of superposition tells us that electric
potential for the system of the point-like charges should be
V  Vi 
i
1
4 0
qi
r
i
(3.4.9)
i
Example 3.4.1: Point charges 4.1 C and 2.2 C are placed on x-axis at (11 m, 0)
and (11m,0) respectively. (a) Sketch the electric potential on the x axis for this system.
(b) Your sketch should show one point on the x-axis between the two charges where the
potential vanishes. Is this point closer to the 4.1 C charge or closer to the 2.2 C
charge? Explain. (c) Find the point referred to in part (b).
Finally, one may notice that since electric field inside of the conductor is always
zero, then equation 3.4.6 shows that change in electric potential has to be equal to zero.
This also means that electric potential inside of the conductor should stay constant. You
will have a chance to observe this in the lab.