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Electrical Energy and Electrical Potential
Two different things that sound alike!
Recall Work: W = F d cos(q)
In order to bring two like charges together work must be done.
In order to separate two opposite charges, work must be done.
As the monkey does + work on the positive
charge against electric force, he increases the
energy of that charge. The closer he brings it,
the more electrical potential energy it has.
This work done by external force against
electrical force is stored as electrical PE, U.
When he let it go, the charge will gain kinetic energy and can do a work.
Try the same thing with grav. force. It is the same!!!! charge → mass
So essentially, potential energy is capacity for doing
work which arises from position or configuration.
Greater amount of charge → greater force needed →
greater work done → greater stored potential energy U.
→ introducing the electrical potential energy per unit charge,
known as electrical potential, which doesn’t depend on
the amount of charge.
If a charge q at point P (in electric field E) has electric
potential energy U, the electric potential V at that point is:
electric potential 
U
V
q
electric potential energy
charge
1J
1V 
1C
The SI unit of electric
potential is the volt.
Because the dependence on the charge q has been divided
out, the electric potential depends only on position.
• Note important difference between energy and potential:
• A point has potential, charge placed there has electric potential energy
+++++
+
+
++++
+
++
+
+ +
+++
Two points that are at the same distance from
the charged object have the same potential.
++ +
++
So, when two charged object are placed
there, they are at the same potential,
but the one with more charge on it has
higher electric potential energy.
The amount of gravitational potential energy depends on the
reference point.
In contrary, we choose that:
The zero of electrical potential and potential energy is at infinity.
Reformulating definition of electric potential:
The potential, V, at a point P in an electric field, is the work done per
unit of positive charge in order to bring it from infinity to that point.
The zero of electrical potential is at infinity.
Following is derivation of potential and
potential energy due to a point charge Q
W = Fs cosθ
W=

 FΔs
i
i
To bring positive charge q to point P from infinity (at constant speed),
one must apply external force |F| = |Fel| to push it against electric field.
r
r


W   Fds  k
r
Qq
Qq
Qq
 Qq 
ds


k
dr

k

k


 r 2
r2
r
r


r
Work done by that force is positive and is stored as potential
energy of charge q at point P.
U=k
Qq
r
and potential at point P is:
V=k
Q
r
Point charge Q - a very confusing picture – time to study!
Electrical potentials can be positive or negative.
F
E
E
F
P
Q
V=k
r
Q
E=k 2
r
Q
E
Q
_
+
Qq
U=k
r
F
P
V=k
E
Q
r
F
F = qE
U=k
Qq
r
E=k
Q
r2
F = qE
Electrical potentials can be positive or negative. V = k
F
F
V2
V1
Q
r
V2
_Q
V1
Q
+
F
V1 < V2
F
V1 > V2
Positive charge accelerates from
higher potential to lower potential.
Negative charge accelerates from
lower to higher potential.
Point charge Q - a very confusing picture – time to study!
Electrical potentials can be positive or negative.
Electrical potentials can be positive or negative. V = k
V1 < V2
V1 > V2
Positive charge accelerates from
higher potential to lower potential.
Negative charge accelerates from
lower to higher potential.
Q
r
Potential Difference Between Two Points (ΔV = VB – VA)
The difference between the potentials at two
different points (A & B) measures the work
done per unit of positive charge in order
to move it from one point to the other.
= ΔU
ΔV = W
q
q
Law of conservation of energy:
change in potential energy = change in kinetic energy
Work – kinetic energy theorem
work done by net force (electric force) = change in kinetic energy
If a charge, q, is moved through a potential difference, ∆V,
then the work done on it is equal to the change in its electric
potential energy which is converted into kinetic energy:
W = DU = q DV = ½ mv2
electron-Volt (eV)
How much work is done in moving one electron
through a potential difference of one volt?
An electron volt is the amount of energy/work it takes to
move an electron through a potential difference of 1 volt.
DU = W = q DV
1 eV  DU = W = (1.6x10-19 C) (1V)
1 eV = 1.6x10-19 J
The electron volt is not a smaller unit for volts!!!
It is a smaller unit for energy.
Uniform electric field, E
A ball accelerates as it loses
potential energy.
Similarly, a positive charge accelerates
from a region of higher potential toward
a region of lower potential.
◊ The change in potential energy of a
charge q is converted into kinetic energy:
∆U = q ∆V = ½ mv2
◊ The change in kinetic energy is equal to the
work done on charge q by el. field:
Fd = qEd = ½ mv2
◊
relationship between uniform electric field E and potential
difference between two points distance d from each other along
electric field line:
ΔV = Ed
→ E = ΔV/d
and NC-1 = Vm-1
For things like batteries, we specify the potential difference between the
contacts (poles) on the batery. So a "D-cell" has a rating of 1.5 volts
which means that every 1C of charge (electrons) that moves from the
negative side of the cell to the positive side will do 1.5 Joules work.
1. Potential difference = 1.5 V →
2. Potential energy of 1 C electrons relative to positive
pole is 1.5 J.
3. That energy is being converted into kinetic energy on
the way to positive pole.
4. Kinetic energy is being converted into work due to
collisions with atoms within conductor. That is what
makes bulb (and wires unfortunately too) to heat up
and give off light (not wires, hopefuly)
5. Electrons reach positive pole completely exausted without
any energy, ready to reenergize again in the cell.
The AA-cell may only light a light bulb for 15 minutes while the D-cell may
keep the same bulb lit for several hours (simply more charge).
In the 12 volt car battery, every coulomb of charge that moves from one
side to the other does 12 Joules worth of work.
In a 120 volt electrical outlet, every Coulomb of charge does 120 Joules
worth of work as it moves from one side of the outlet to the other.
surprise, surprise ! We use the same name for different things, and even
worse we use couple of different names to express the same thing, like:
1. The variable we use for potential, potential difference, and the unit
for potential difference (volts) is V. Cute!!!!!
2. Don't let that confuse you when you see V = 1.5V
3. Electric potential energy is not the same as electrical potential.
4. The electron volt is not a smaller unit of the volt, it's a smaller unit of
the Joule.
5. Electrical potential can also be described by the terms, potential
difference, voltage, potential drop, potential rise, electromotive force,
and EMF. These terms may differ slightly in meaning depending on
the situation.