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Electrical Potential
Young and Freedman
Chapter 23
Work
It takes work to lift a mass in a GRAVITATIONAL FIELD
The work done increases the Gravitational Potential Energy
Work
q
q
Q
It takes work to lift a charge in an electric field
This increases the charges Electrical Potential Energy
Review: Work, Energy, Potential Energy, Conservation of Energy
v r
Unit are Newtons · Meters = Joules
!W = F " !x
e
nc
sta
Di
e
rc
Fo
e
on
kd
or
W
Wa #b = !
b
a
v r
b
F " dl = ! F cos $dl
a
Forces like Gravity and the Electric Force are Conservative Forces
[i.e. No Friction]. Work done by conservative forces can be described
as a change in Potential Energy
Wa #b = (U a " U b ) = "(U b " U a ) = " !U
ΔU can be positive or negative
ΔU positive – Work is “done” in moving from point a to point b
ΔU negative – Energy is “released” in moving from point a to point b
To avoid confusion it is often helpful to think about a ball rolling up or down a hill
ΔU positive – Work is “done” in rolling the ball UP the hill
– higher up means higher EP
ΔU negative – Energy is “released” when the ball rolls down
- lower down means lower EP
Electrostatic Potential Energy in a Uniform Electric Field
+
b
yb
ya
r
E
a
+ q
y
r
E = ! Eyˆ
constant
-
r
Move a positive charge in a vertical direction dl = yˆ dl
r
b v
b
b
W = $ F # dl = $ " qEyˆ # yˆ dl = $ " qEdl = " qE ( yb " ya ) = " qE!y
a
a
a
W = " !U
Looks like U gravity = mg!h
!U a "b = qE!y
b
+
c
Now move from b → c
+
r
E
a
-
W =!
y
c
v r
F " dl
b
r
r
But F is perpendicular to dl
So no work is done and there is
no change in U
!U a "c = qE!y
Now, instead, go directly from a → c and compute ΔU
c
b
Δy
a
+
φ
L
+
r
E
-
W =!
c
a
y
v r
c
F " dl = ! F cos(# )dl
a
c
W = " qE cos(# ) ! dl
a
W = " qE cos(! ) L
Path length is given by:
y = L cos(! ) # L =
"y
cos(! )
Therefore: W = " qE!y
Work done on a charge in an electric Field is INDEPENDENT OF PATH
Since
Wa #b = (U a " U b ) = "(U b " U a ) = " !U is independent of path,
all that matters are the values of the potential energy at the start and
at the end
Work and Potential Energy – Keeping the signs straight
Wa #b = !
b
a
r v b
F " dl = ! F cos $dl
Work done by a Force
a
+
+
+
+
-
-
-
-
Work and Potential Energy
Wa #b = !
b
a
r v b
F " dl = ! F cos $dl
Work done by a Force
a
The sign is critically important
r
r
F = qE
r
F" ME"
Direction of Motion
The Work in the equations above refers to the
Work DONE BY THE ELECTRIC FIELD
If I slowly move the charge,* I will always be
applying a force which is equal and opposite to the
Electrostatic Force. The work DONE BY ME will
be equal and opposite to the work done by the
E field.
* no change in kinetic energy
The path of integration does NOT matter for a
“conservative” force such as the Electrostatic Force
Wa #b = !
b
a
r v b
F " dl = ! F cos $dl
a
Example: Electric potential energy of two
point charges
Fr =
b
b
a
a
Wa )b = ( Fr dr = (
qq0
qq0
dr =
2
4*+ 0 r
4*+ 0
(
b
a
qq0
4!" 0 r 2
qq0 & 1 1 #
1
$$ ' !!
dr =
2
4*+ 0 % ra rb "
0r
The path of integration does NOT matter
for “conservative” forces such as the
electrostatic force
r v b
Wa #b = ! F " dl = ! F cos $dl
a
a
qq0 & 1 1 #
$$ ' !!
Wa (b = U a ' U b = '+U =
4)* 0 % ra rb "
b
NO ABSOLUTE ZERO for potential
energy
• It is the CHANGE in potential energy that is important
• We can pick any distance to be the “baseline” for
potential energy
• For convenience we pick the zero of potential energy
to refer to the Energy of the system when the
charges are VERY far away from one another
“infinity”
m
h
M
We pick ZERO for potential energy to be “Infinite Separation”
qq0
' *U =
4() 0
qq0
' *U =
4() 0
&1 1#
$$ ' !!
% ra rb "
&
1#
$$ 0 ' !!
rb "
%
Let ra = ∞
Define U (r ) = qq0 1
4!" 0 r
The Potential Energy:
qq0 1
U (r ) =
4!" 0 r
is related to the amount of work that must be done to bring
two charges together from far away.
I must do POSITIVE work to
bring two like charges together
I must do NEGATIVE work to
bring two like charges together
Electrostatic potential energy for several
point charges
What is the electrostatic potential
energy of q0 due to q1, q2, and q3 ?
Electrostatic Potential Energy is a SCALAR and adds algebraically
This is different from the Electric Field; a VECTOR that adds vectorially
Problem: A system of point charges
q1 $ q2 q3 ' *e $ e e '
U=
& + )=
& + )
4 "# 0 % a 2a ( 4 "# 0 % a 2a (
*e2 $ 3 '
U=
& )
4 "# 0 % 2a (
Check sign - Sign is negative, which is correct for
opposite charges. The system is at a lower potential
energy than if q1 was at infinity
!
Problem: A system of point charges
U
q1 $ q2 q3 ' *e $ e e '
U =q $ q& +q ') = e &$ *e
+ )e '
24 "# 0 %1 a
2 ( 4 "# 0 % a 2a (
2a
=
& + )=
& + )=0
2
4 "#*e
$ 3 'a (
0 % a
U=
& )
4 "# 0 % 2a (
4 "# 0 % a
a(
Check sign - Sign is negative, which is correct for
opposite charges. The system is at a lower potential
energy than if q1 was at infinity
!
Electrostatic potential
The Electrostatic Potential Energy of a point charge q0
and a collection of Charges qi :
Sample Problem #3
q1
q2
q3
What is the TOTAL potential energy of this charge distribution?
Start by removing ALL charges to ∞.
This corresponds to the “zero” of Potential Energy
Step 1: All charges at ∞
U=0
Step 2: Bring Charge q3in from ∞ . This requires no work
because there are no other charges present.
U=0
Step 3: Bring charge q2 in from infinity
q1
1
U=
4'( 0
q3
q2
& q 2 q3 #
$$
!!
% r23 "
Step 4: Bring charge q1 in from infinity
q1
q2
1
U=
4'( 0
q3
& q2 q3 q1q2 q1q3 #
$$
!!
+
+
r12
r13 "
% r23
The electrostatic potential energy of
collection of point charges
1
U=
4"# 0
!
i< j
qi q j
rij
Note: BE CAREFUL NOT TO DOUBLE COUNT
Electrostatic Force Electrostatic Field
(Chapter 21)
r
1 qi
F (r ) = q !
r̂
2 i
i 4"# 0 ri
v
1 qi
(
)
E r =!
r̂
2 i
i 4"# 0 ri
We defined the concept of the Electric Field as
a generalization of the Electric Force.
The Electric Field assigns a VECTOR to each point in Space
Electrostatic Potential Energy
of a charge q0 and a collection of charges qi
q0
U=
4() 0
' q1 q2 q3
$
q0
%% + + + ..."" =
& r1 r2 r3
# 4() 0
qi
!i r
i
If the charges qi remain fixed and I move q0, I can express
Potential Energy as a Function of the position of q0
q0
U (r ) =
4"# 0
qi
!i r
i
Electrostatic Potential
1 q
U=
4 "# 0 r
U
1
qi
V= =
$
q0 4 "# 0 i ri
Potential of a
set of POINT CHARGES
Potential energy due to a
POINT CHARGE
!
IMPORTANT: Like Work!-Electrostatic Potential is
a “relative” quantity.
r r
Wa"b = U a #U b = $ a F % dl =
b
divide by q0
!
Va # Vb = !
b
a
r r
$ a q0 E % d l
b
r r b
E " dl = ! E cos $dl
a
UNITS : Volt = Joule/Coulomb
ΔV = 1.5 volts
Electrostatic Potential Energy 
Electrostatic Potential
q0
U (r ) =
4"# 0
qi
!i r
i
1
V (r ) =
4"# 0
qi
!i r
i
We defined the concept of the Electric Potential as
a generalization of the Electric Potential Energy.
The Electric Potential assigns a Scalar to each point in Space
Finding potential by integration
Find the potential at a distance r from a point charge using
integration
Va # Vb = !
b
a
r r b
E " dl = ! E cos $dl
a
reduces to
V "0 =
$
#
r
Edr =
q
="
4%&0 r
q
V=
4 %& 0 r
$
#
r
q
dr
2
4 %& 0 r
#
'
q *
= 0 " )"
,
( 4 %& 0 r +
r
Example 23.9: Potential of oppositely charged parallel plates
Find the potential at any height y between two
oppositely charged parallel plates
Va # Vb = !
b
a
V (y ) =
r r b
E " dl = ! E cos $dl
a
U (y ) q0 Ey
=
= Ey
q0
q0
We have chosen V=0 at point b
Example 23.8: A charged conducting sphere
A solid conducting sphere of radius R has a total charge q.
Find the potential both outside and inside the sphere.
E is 0 everywhere within the sphere,
else charge would move to make it
zero. Hence no work is done on a test
charge moving around within the sphere
and the potential is the same at every
point within the sphere. The value of
the potential is defined by the surface of
the sphere.
NOTE: A conducting sphere is
always an equipotential surface
Equipotential Surfaces and Electric Field Lines
Equipotential Surfaces
are always perpendicular
to Electric Field Lines
Chapter 23 Summary
Chapter 23 Summary cont.
End of Chapter 23
You are responsible for the material covered in T&F Sections 23.1-23.4
You are expected to:
• Understand the following terms:
Work, Potential Energy, Electrostatic Potential, Potential Difference,
Equipotential Surface
•
Understand the relationship between work and potential energy. In particular, be
able to distinguish between work done “by the field” and work done by the force
moving a charge in a field.
•
Be able to calculate the electrostatic potential energy of simple charge
geometries using either the sum rule for discrete geometries or the integral rule
for continuous geometries
•
CLEARLY DISTINGUISH BETWEEN POTENTIAL ENERGY AND
ELECTROSTATIC POTENTIAL involved.
•
Be able to calculate the electrostatic potential at a point in space due to simple
charge geometries using either the sum rule for discrete geometries or the
integral rule for continuous geometries
•
Understand the convention of setting the zero of potential energy at “infinity”
Recommended Y&F Exercises chapter 23:
• 6,9,14,21,23,35,40