Chapter 5 Work Done by the Electric Field, and, the Electric Potential
5 Work Done by the Electric Field, and, the Electric Potential
When a charged particle moves from one position in an electric field to another position in that
same electric field, the electric field does work on the particle. The work done is conservative;
hence, we can define a potential energy for the case of the force exerted by an electric field. This
allows us to use the concepts of work, energy, and the conservation of energy, in the analysis of
physical processes involving charged particles and electric fields.
We have defined the work done on a particle by a force, to be the force-along-the-path times the
length of the path, with the stipulation that when the component of the force along the path is
different on different segments of the path, one has to divide up the path into segments on each
of which the force-along-the-path has one value for the whole segment, calculate the work done
on each segment, and add up the results.
Let’s investigate the work done by the electric field on a charged particle as it moves in the
electric field in the rather simple case of a uniform electric field. For instance, let’s calculate the
work done on a positively-charged particle of charge q as it moves from point P1 to point P3
P2
b
P3
c
a
E
P1
along the path: “From P1 straight to point P2 and from there, straight to P3.” Note that we are
not told what it is that makes the particle move. We don’t care about that in this problem.
Perhaps the charged particle is on the end of a quartz rod (quartz is a good insulator) and a
person who is holding the rod by the other end moves the rod so the charged particle moves as
specified.
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
Along the first part of the path, from P1 to P2 , the force on the charged particle is perpendicular
to the path.
b
P2
P3
F
c
a
E
P1
The force has no component along the path so it does no work on the charged particle at all as
the charged particle moves from point P1 to point P2.
W12 = 0
From P2, the particle goes straight to P3.
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
On that segment of the path (from P2 to P3 ) the force is in exactly the same direction as the
direction in which the particle is going.
P2
F
b
P3
c
a
E
P1
As such, the work is just the magnitude of the force times the length of the path segment:
W23 = F b
The magnitude of the force is the charge of the particle times the magnitude of the electric field
F = qE, so,
W23 = q E b
Thus, the work done on the charged particle by the electric field, as the particle moves from
point P1 to P3 along the specified path is
W123 = W12 + W23
W123 = 0 + q E b
W123 = q E b
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(This is just an answer to a sample
problem. Don’t use it as a starting point
for the solution to a homework or test
problem.)
Chapter 5 Work Done by the Electric Field, and, the Electric Potential
Now let’s calculate the work done on the charged particle if it undergoes the same displacement
(from P1 to P3 ) but does so by moving along the direct path, straight from P1 to P3.
b
P2
P3
c
a
θ
F
E
P1
The force on a positively-charged particle being in the same direction as the electric field, the
force vector makes an angle θ with the path direction and the expression
W = F ⋅ ∆r
for the work becomes
W13 = F c cos θ
W13 = q E c cos θ
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
Analyzing the shaded triangle in the following diagram:
b
P2
θ
P3
c
a
θ
F
E
P1
we find that cos θ =
b
. Substituting this into our expression for the work ( W13 = qE c cos θ )
c
yields
W13 = q E c
W13 = q E b
b
c
(This is just an answer
to a sample problem.)
This is the same result we got for the work done on the charged particle by the electric field as
the particle moved between the same two points (from P1 to P3 ) along the other path (P1 to P2 to
P3 ). As it turns out, the work done is the same no matter what path the particle takes on its way
from P1 to P3. I don’t want to take the time to prove that here but I would like to investigate one
more path (not so much to get the result, but rather, to review an important point about how to
calculate work). Referring to the diagram:
36
Chapter 5 Work Done by the Electric Field, and, the Electric Potential
P2
b
P3
d
P5
a
a
E
P1
b
d
P4
Let’s calculate the work done on a particle with charge q, by the electric field, as the particle
moves from P1 to P3 along the path “from P1 straight to P4 , from P4 straight to P5 , and from P5
straight to P3.” On P1 to P4 , the force is in the exact same direction as the direction in which
the particle moves along the path, so,
W14 = F (b + d )
W14 = qE (b + d )
From point P4 to P5 , the force exerted on the charged particle by the electric field is at right
angles to the path, so, the force does no work on the charged particle on segment P4 to P5 .
W45 = 0
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
On the segment from P5 to P3,
d
P2
P3
P5
F
a
a
E
P1
b
d
P4
the force is in the exact opposite direction to the direction in which the particle moves. This
means that the work done by the force of the electric field on the charged particle as the particle
moves form P5 to P3 is the negative of the magnitude of the force times the length of the path
segment. Thus
W53 = − F d
W53 = − qE d
and
W1453 = W14 + W45 + W53
W1453 = qE (b + d ) + 0 + (− q E d )
W1453 = q E b
(This is just an answer
to a sample problem.)
As advertised, we obtain the same result for the work done on the particle as it moves from P1 to
P3 along “P1 to P4 to P5 to P3” as we did on the other two paths.
Whenever the work done on a particle by a force acting on that particle, when that particle moves
from point P1 to point P3 , is the same no matter what path the particle takes on the way from P1
to P3, we can define a potential energy function for the force. The potential energy function is an
assignment of a value of potential energy to every point in space. Such an assignment allows us
to calculate the work done on the particle by the force when the particle moves from point P1 to
point P3 simply by subtracting the value of the potential energy of the particle at P1 from the
value of the potential energy of the particle at P3 and taking the negative of the result. In other
words, the work done on the particle by the force of the electric field when the particle goes from
one point to another is just the negative of the change in the potential energy of the particle.
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
In determining the potential energy function for the case of a particle of charge q in a uniform
electric field E , (an infinite set of vectors, each pointing in one and the same direction and each
having one and the same magnitude E ) we rely heavily on your understanding of the nearearth’s-surface gravitational potential energy. Near the surface of the earth, we said back in
volume 1 of this book, there is a uniform gravitational field, (a force-per-mass vector field ) in
the downward direction. A particle of mass m in that field has a force “mg downward” exerted
upon it at any location in the vicinity of the surface of the earth. For that case, the potential
energy of a particle of mass m is given by mgy where mg is the magnitude of the downward
force and y is the height that the particle is above an arbitrarily-chosen reference level. For ease
of comparison with the case of the electric field, we now describe the reference level for
gravitational potential energy as a plane, perpendicular to the gravitational field g , the force-permass vector field; and; we call the variable y the “upfield” distance (the distance in the direction
opposite that of the gravitational field) that the particle is from the reference plane. (So, we’re
calling the direction in which the gravitational field points, the direction you know to be
downward, the “downfield” direction.)
Now let’s switch over to the case of the uniform electric field. As in the case of the near-earth’s
surface gravitational field, the force exerted on its victim by a uniform electric field has one and
the same magnitude and direction at any point in space. Of course, in the electric field case, the
force is qE rather than mg and the characteristic of the victim that matters is the charge q rather
than the mass m. We call the direction in which the electric field points, the “downfield”
direction, and the opposite direction, the “upfield” direction. Now we arbitrarily define a plane
that is perpendicular to the electric field to be the reference plane for the electric potential energy
of a particle of charge q in the electric field. If we call d the distance that the charged particle is
away from the plane in the upfield direction, then the potential energy of the particle with charge
q is given by
U = q Ed
where:
U is the electric potential energy of the charged particle,
q is the charge of the particle,
E is the magnitude of every electric field vector making up the uniform electric field, and
d is the “upfield” distance that the particle is from the U = 0 reference plane.
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
Let’s make sure this expression for the potential energy function gives the result we obtained
previously for the work done on a particle with charge q, by the uniform electric field depicted in
the following diagram, when the particle moves from P1 to P3
P3
E
P1
b
Positive direction for measurements of d, the upfield
distance that a point is from the reference plane.
The reference plane
viewed edge on.
All points on this
plane are at d = 0.
As you can see, I have chosen (for my own convenience) to define the reference plane to be at
the most downfield position relevant to the problem. With that choice, the particle of charge q,
when it is at P1 has potential energy qEb (since point P1 is a distance b “upfield” from the
reference plane) and, when it is at P3, the particle of charge q has potential energy 0 since P3 is
on the reference plane.
W13 = − ∆U
W13 = −(U 3 − U 1 )
W13 = −(0 − qEb)
W13 = q E b
(This is just an answer to a sample problem.)
This is indeed the result we got (for the work done by the electric field on the particle with
charge q as that particle was moved from P1 to P3 ) the other three ways that we calculated this
work.
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Chapter 5 Work Done by the Electric Field, and, the Electric Potential
The Electric Potential Energy per Charge
The expression for the work that we found above had the form “the charge of the victim times
other stuff.” Likewise, the potential energy of the victim (see above) has the form “the charge of
the victim times other stuff.” In both cases the “other stuff” consisted of quantities
characterizing the electric field and positions in space. This turns out to be a general result: The
electric potential energy of a charged particle (victim) in any electric field (not just a uniform
electric field ) can be expressed as the product of the charge of the victim, and, quantities used to
characterize the electric field in the region of space in which the particle finds itself. As such, we
can always divide the potential energy of the victim by the charge of the victim to obtain what
can be called the electric potential energy per charge for the point in space at which the victim
finds itself. No matter what the charge of the victim is, the potential energy of the victim divided
by the charge of the victim always yields the same value for the potential-energy-per-charge-ofwould-be-victim. This is because the potential-energy-per-charge-of-would-be-victim is a
characteristic of the point in space at which the victim finds itself, not a characteristic of the
victim. This means that we can specify values of potential-energy-per-charge-of-would-bevictim (which we will use the symbol ϕ to represent) for all the points in a region of space in
which there is an electric field, without even having a victim in mind. Then, once you find a
victim, the potential energy of the victim at a particular point in space is just
U = qϕ
(5-1)
where:
U is the electric potential energy of the victim (the charged particle in the electric field ),
q is the charge of the victim, and,
ϕ is the electric-potential-energy-per-charge-of-would-be-victim (also known more simply
as the electric potential ) of the point in space at which the victim finds itself.
Okay, I spilled the beans in the variable list; “potential-energy-per-charge-of-would-be-victim” is
just too much of a mouthful so we call it the electric potential. Now, for the potential energy U
to come out in Joules in the expression U = qϕ, with q having units of coulombs, the electric
potential ϕ must have units of J/C. The concept of electric potential is such an important one that
we give its combination unit (J/C) a name. The name of the unit is the volt, abbreviated V.
1 volt = 1
joule
J
or 1V = 1
coulomb
C
For the case of a uniform electric field, our expression U = q E d for the electric potential energy
of a victim with charge q, upon division by q, yields, for the electric potential at a point of
interest in a uniform electric field,
ϕ = Ed
(5-2)
where:
ϕ is the electric potential at the point of interest,
E is the magnitude of every electric field vector in the region of space where the uniform
electric field exists, and,
d is the upfield distance that the point of interest is from the (arbitrarily-chosen) reference
plane.
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