Electric Fields and Electrical Potential Energy
Electric Field
There is an electric field around any charged object. In other words, objects with charge (or capable of having an
induced charge) brought near a charge object will experience a force.
You can visualize an electric field by playing around with something called a test charge. A test charge is what it
sounds like: you move it around in an electric field to “test” what the object would feel.
About a test charge:



Very small (won’t alter the field)
Positive (Why? Because we said so, that’s why….its just a convention, totally arbitrary)
It ultimately won’t matter
+
Q
q
We use the test charge to help us draw electric field lines. This is a way of visualizing the completely invisible
electric fields.
Practice: Drawing an Electric Field
Q
Rules for Electric Field Lines:


They show the direction of the force a point charge would feel. Remember, point charges are positive, so
electrons or negative charges would experience the opposite force.
The more lines there are in an area, the stronger the field is there.
Practice: Drawing the Electric Field on Multiple Charges
Q
Story Notes, created with materials from Knapp notes
Calculating Electric Field Intensity
If you want to calculate the electric field strength around an object, first consider force acting between an charge, Q,
and its point charge q0.
𝐹=
𝑘𝑄𝑞𝑜
𝑑2
The test charge doesn’t really matter though, does it? We just used it to test the strength of the field around the
charge Q. So we can divide it out.
𝐸=
𝐹
𝑘𝑄𝑞𝑜
=
𝑞0 𝑞0 𝑑2
𝐸=
𝐹
𝑘𝑄
= 2
𝑞𝑜 𝑑
The units for electric field strength are N/C.
Memorize This!
𝐸=
𝐹
𝑘𝑄
= 2
𝑞𝑜 𝑑
AP Practice: A really easy question
An electron is accelerated from rest for a time of 10–9 second by a uniform electric field that exerts a force of
8.0 x 10–15 newton on the electron.What is the magnitude of the electric field?
(A) 8.0 × 10–24 N/C (B) 9.1 × 10–22 N/C (C) 8.0 × 10–6 N/C (D) 2.0 × 10–5 N/C (E) 5.0 × 104 N/C
Story Notes, created with materials from Knapp notes
AP Practice: A little bit harder now
Charges –Q and +Q are located on the x– and y–axes, respectively, each at a distance d from the origin O, as
shown above.
Electrical Potential Energy
Review from Mechanics
Potential energy from gravity is
U g  mgh .
Work takes place when a force acts over a distance.
W Fr
The unit for work is the joule (J). A joule is a newton  meter.
The change in potential energy does not depend on the height, it depends on the change in
height.
Story Notes, created with materials from Knapp notes
Fg
increase in
gravitational
potential
energy
We have a comparable situation to that of gravity when we have two opposite charges. We have a large charge Q.
Placed near this charge is a small test charge q. As we increase the distance between q and Q, we have to do work
to move the charge and we increase its potential energy .
FE
+
+
increase in
electric
potential
energy
-
-
FE
-
FE
-
-
+
Unlike charges
Force is attractive
increase in
electric
potential
energy
-
+ F
E
+
+
Like charges
Force is repulsive
Deriving the Equation for Electric Potential
We know that potential energy is force times distance.
U  Fr
The force acting between the electron and the proton is:
F
1 q1q2
4 0 r 2
We can plug in Coulomb’s law for the force in the potential energy equation.
Memorize This!
 1 q1q2 
U 
r
2 
4


r
0



1 q1q2
4 0 r
So there it is. An equation for electrical potential energy:
Practice Question: Introduction to Potential Energy
An electron in a hydrogen atom is at its lowest energy state. It is at a distance of 5.29 x 10-11 m from the proton in
the nucleus. What is its potential energy relative to the proton?
Important: Potential Energy and Work are scalars. When we dealt with electric field strength and force, we
were dealing with vectors, so direction mattered and magnitudes needed to be added as scalars. Because potential
energy and work (which we will discuss in a minute) are scalars, we do not need to worry about a direction. We can
simply add their magnitudes when adding them.
Story Notes, created with materials from Knapp notes
AP Practice Question: Scalars and Vectors
Work in Electric Fields
Changes in the electric field = Work
Just like in a gravitational field, you do work by moving the object closer or further away.
Unlike charges
Pull apart
Push together
Pull apart
Push together
Like charges
Deriving the Equation for Work
We know that work is:
W Fr
The force exerted on a test charge in an electric field is:
E
F
q
F  qE
We now put these two equations together and we get:
W  Fd
  qE  d
 qEd
Story Notes, created with materials from Knapp notes
+W
-W
-W
+W
Increase potential
Decrease potential
Decrease potential
Increase
This gives us an equation for the work done in an electric field in moving a charge form one position to another:
W is the work done in Joules, q is the charge of the test charge (not the charge making the field), E is the electric field strength, and
d is the displacement of the charge.
Dealing with Signs
This is awkward. Generally, when dealing with work, we’ve thought about things from our perspective- the work
we as an outside force need to do. In order to get your signs right with electrostatics, you need to think about work
from the field’s perspective.
Memorize This!
General Rule: The field does positive work a charge in order to
decrease its potential energy.
That means that if you use the equation above, it will give you the work the field does
in moving a particle.
If you are asked about the work an outside force does, you will need to use the
opposite sign.
Scenario 1
Let’s talk through a scenario. Let’s say the charge in the scenario on the left is
positive and it is moving from Point A to Point B.
E
We deal with the electric field strength’s direction by putting a direction on d.
A
B
If you move in the direction of the electric field, d is
positive.
d
So, using W=qEd, the field does positive work. The particle was moved
toward a place of lower potential energy. So we can say:
For this scenario, where a positive charge is moved from A to B:
q
E
d
W
∆U
+
Deal with this with d
+
+
-
The equation only works if you state work in terms of the work done by the
field.
Story Notes, created with materials from Knapp notes
Memorize This!
Scenario 2
Now, let’s look at the same field with a negative charge.
The charge would naturally be pushed which direction? Remember that the field
lines were made using a positive charge.
E
A
B
If the charge moved from Point A to Point B, is the field doing positive or
negative work? Use the equation W=qEd
The ∆U would therefore be
charge reaches Point B it has ______________ potential energy.
. So when the
d
Scenario 3
Now, think about it from the perspective of an outside force. Imagine the charge is negative.
The charge would naturally want to go in which direction? Remember that the
field lines were made using a positive charge.
E
A
B
If you (as an external force) moved the charge from Point A to Point B, would
you have to do positive or negative work? Use W=qEd.
d
The ∆U would therefore be
______________ potential energy.
. So when the charge reaches Point B it has
Potential Difference
Let’s say we didn’t want to look at the potential energy of a specific particle, but wanted to look at the difference in
potential between two given points. This is called the potential difference.
POTENTIAL DIFFERENCE  CHANGE IN POTENTIAL ENERGY
PER CHARGE
The symbol for potential difference is V or V.
Deriving the Equation for Potential Difference
V  Potential Difference
V
U
q
Story Notes, created with materials from Knapp notes
Memorize This!
If we look at the equation for potential energy and divide both sides by the charge q, then we get the potential
difference.
U  qEd
U  qEd

q
q
We divide both sides by q (which is mathematically legal).
V   Ed
This equation is commonly rearranged to this form:
Memorize This!
This is the equation that you will have available for the AP Physics Test. Notice also that the
electric field can be expressed in units of V/m as well as N/C.
The unit for potential difference is the volt (V). The volt is a joule/Coulomb.
1V1
J
C
A potential difference of one volt means that it takes one joule of energy to move a one Coulomb charge in the field. A potential
difference of 1000 V means that it would take 1000 J to move 1 C of charge in the field.
Potential difference is also called electric potential or voltage.
The Equations You Get on the AP Test:
E
F
q
U E  qV 
1 q1q2
4 0 r
E Avg  
V
d
Practice Question: Working with Potential
The potential difference between two points is 12.0 V. What amount of work is needed to move a 2.00 C
charge within the field?
AP Question: Work and Voltage
The work that must be done by an external agent to move a point charge of 2 mC from the origin to a point 3 m
away is 5 J. What is the potential difference between the two points?
(A) 4 × 10–4 V (B) 10–2 V (C) 2.5 × 103 V (D) 2 × 106 V (E) 6 × 106 V
Story Notes, created with materials from Knapp notes
Practice Question: Electric Potential
Two charged plates are 5.00 cm apart. The electric field between them is 775 N/C. What is (a) the potential
difference between the plates and (b) what work is done moving an electron from one plate to another?
Story Notes, created with materials from Knapp notes