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Q22.3
A conducting spherical shell with inner radius a and
outer radius b has a positive point charge Q located at
its center. The total charge on the shell is –3Q, and it
is insulated from its surroundings. In the region a < r
< b,
A. the electric field points radially outward.
B. the electric field points radially inward.
C. is zero.
D. not enough information given to decide
Q22.5
There is a negative surface charge density in a certain region on the surface of a solid
conductor.
Just beneath the surface of this region, the electric field
A. points outward, toward the surface of the conductor.
B. points inward, away from the surface of the conductor.
C. points parallel to the surface.
D. is zero.
E. not enough information given to decide
Q22.6
For which of the following charge distributions would Gauss s law not be
useful for calculating the electric field?
A. a uniformly charged sphere of radius R
B. a spherical shell of radius R with charge uniformly distributed over its
surface
C. a right circular cylinder of radius R and height h with charge uniformly
distributed over its surface
D. an infinitely long circular cylinder of radius R with charge uniformly
distributed over its surface
E. Gauss s law would be useful for finding the electric field in all of these
cases.
Summary: Gauss’s Law
• Electric Flux: flow of electric field through a surface
• 
(Just like the light flow from your flashlight)
• Gauss’s Law: total electric flux through a closed surface (Gaussian
Surface) is porportional to the total charge enclosed.
"E =
#
!
! q
E • dA =
$0
• Calculating Electric Field using Gauss’s law:
• 
!
useful when charge distribution is highly symmetric
• Special properties of a conductor:
• 
Field within an electrostatic conductor is always 0
• 
The charges are distributed on the surface (not inside)
•  What’s real: Gaussian Surface/Electric Flux/Electric Field
Chapter 23: Electric Potential
•  Electric Potential Energy: When electric force
does work, what happens?
•  New Concept: Electric Potential (Volts)
•  1.5Volt Battery, 110 Volt Power line; What
does it mean?
•  equipotential surfaces: regions of equal
potential
•  Calculate electric field from electric potential
Introduction
•  Why does water flow from
high to low?
•  What happens if a charge is
placed in an electric field?
Does it gain kinetic energy?
Where does this energy come
from?
•  Welding: electrons coming
from the welder rod to the
material. Why? If the rod is
far away from the material,
what will happen?
Announcement:
Volunteer needed for Thursday
(small person wanted ;wear as
few metal objects as possible)
Electric force is a conservative force
•  The work done raising a basketball against gravity
depends only on the potential energy, how high the
ball goes. It does not depend the path (the definition
!
of a conservative force)
b !
Wa!>b =
"
a
F • dl = !#U = !(U b !U a )
•  A point charge moving in a field exhibits similar
behavior (The field does not have to be uniform).
U = qEy
d = ya ! yb
Only the displacement in
y direction matters!
Wa!>b = qEd = !"U = !(U b !U a ) = "K
How fast will the charge move at b?
An electric charge moving in an electric field
Positve Charge
Negative Charge
In each case, how does
the electric potential energy
change when moved from
point a to point b?
Moved along the electric
force, U decreases!
If moved opposite to the
force, U increases!
When a positive charge moves in the direction of the
electric field,
A. the field does positive work on it and the
potential energy increases.
B. the field does positive work on it and the
potential energy decreases.
C. the field does negative work on it and the
potential energy increases.
D. the field does negative work on it and the
potential energy decreases.
!
E
+q
Motion
!
E
When a positive charge moves opposite to the
direction of the electric field,
A. the field does positive work on it and the
potential energy increases.
B. the field does positive work on it and the
potential energy decreases.
C. the field does negative work on it and the
potential energy increases.
D. the field does negative work on it and the
potential energy decreases.
!
E
Motion
+q
!
E
When a negative charge moves in the direction of
the electric field,
A. the field does positive work on it and the
potential energy increases.
B. the field does positive work on it and the
potential energy decreases.
C. the field does negative work on it and the
potential energy increases.
D. the field does negative work on it and the
potential energy decreases.
!
E
–q
Motion
!
E
When a negative charge moves opposite to the
direction of the electric field,
A. the field does positive work on it and the
potential energy increases.
B. the field does positive work on it and the
potential energy decreases.
C. the field does negative work on it and the
potential energy increases.
D. the field does negative work on it and the
potential energy decreases.
!
E
Motion
–q
!
E
Electric Potential energy of two point charges
•  In a uniform field, U=qEy
•  What if the field is not uniform: Field of a point charge
1 qq0
F=
4 πε 0 r 2
b 1
qq0
qq0 1
1
W a −>b = ∫ a
dr
=
(
−
)
2
4 πε 0 r
4 πε 0 r a rb
If we define U=0 at rinfinity
U=
1 qq0
4 πε 0 r
What if the path from a is not a straight line?
€
Electric Potential Energy of two point charges
•  The path does not matter!
•  Here is the proof!
Wa!>b =
Wa!>b =
"
"
b
a
b
a
! !
F • dl =
"
b
a
F cos ! dl
1 qq0
cos ! dl
2
4"# 0 r
cos ! dl # dr!
1 qq0
U=
4!" 0 r
It DEPENDS on the signs!
Potential energy curves—PE versus r
•  We usually define the potential
energy at infinity to be zero!
•  Same sign charges: Always
positive PE.
•  Opposite sign charges?
Example: A position (+e) is moving around an alpha particle (+2e).
At r=0.1nm, v=3.00x106m/s, moving
Directly away. What is the speed at r=0.2nm?
What if it is very very far away?
What if it’s an electron?
!K = "!U
K b " K a = U a "U b =
(+e)(+2e) 1
1
( " )
4!" 0
ra rb
1
1
(+e)(+2e) 1
1
mvb2 = mva2 +
( " )
2
2
4!" 0
ra rb
How far can an electron go?
Electrical potential and multiple point charges
•  The electric potential energy: It is not a vector!
•  The electric potential energy between multiple charges is the
algebraic sum of that associated with all charges
q0
qi
U=
!
4!" 0 i ri
What about the PE between all of them: what’s needed to
Assemble these charges together?
qi q j
1
U=
!
4!" 0 i< j rij
Two charges : q1=-e at x=0, and
q2=+e at x=a.
How much work is needed to bring
q3=+e from infinity to x=2a?
What’s the total potential energy of
the three charge system?
A few words on electric potential energy
We can define an electric potential energy because electric force is a
conservative force
We usually define the potential energy at infinite to be zero
Typically, it’s the difference of potential energies between two
points that matter!
While the electric field does not depend on the sign/magnitude of
the test charge, the electric potential energy does depend on the
charges!
Q23.5
Charge #2
The electric potential energy of two point charges
approaches zero as the two point charges move
farther away from each other.
If the three point charges shown here lie at the
vertices of an equilateral triangle, the electric
potential energy of the system of three charges is
+q
Charge #1
+q
y
–q
x
A. positive.
C. zero.
B. negative.
D. not enough information given to decide
Charge #3
Q23.6
Charge #2
The electric potential energy of two point charges
approaches zero as the two point charges move
farther away from each other.
If the three point charges shown here lie at the
vertices of an equilateral triangle, the electric
potential energy of the system of three charges is
–q
Charge #1
+q
y
–q
x
A. positive.
C. zero.
B. negative.
D. not enough information given to decide
Charge #3
The electrical potential: What does 1.5Volt battery mean?
•  Electric potential: Electric potential energy per
unit charge
U
q0
V = volt = J /C = joule /coulomb
Vab = Va − Vb = (U a − U b ) /q0
V=
€
•  Vab: The potential of a with respect to b, equal
to the work needed to move a unit charge from
b to a slowly against the electric force, or the
work done by the electric force moving the
unit charge from a to b
•  Voltmeter: Measuring Potential
•  Unit of Electron Volts: It’s an energy unit, not
potential (MeV, GeV, TeV…)
Calculating Electric Potential:
Point Charge
V=
U
1 q
=
q0 4 πε 0 r
Multiple Charge:
V=
U
1
q
=
∑
q0 4 πε 0 i ri
Is this a vector sum
or scalar sum?
Continuous Charges:
U
1
V= =
q0 4 πε 0
∫
dq
1
=
r
4 πε 0
∫
 
ρ ( r )dr
r
If the electric field is known:
Va − Vb =
∫
b
a


b
E • dl = ∫ a E cos φdl
1V/m=1 volt/meter = 1 N/C = 1 newton/coulomb
€
A particle accelerator imparts amazingly large energies
•  What happens if a particle is accelected to
millions/billions, or trillions of ev?
•  What the typical energy of visible light?
What kind of speed will these particles
reach?
Fermi Lab, Illinois
A proton(+e=1.602x10-19C) is accelerated from point a to be (distance
d=0.50m). E=1.5x107 V/m=1.5x107 N/C (uniform)
What is the work done by the field?
Wa!>b = Fd = (qE)d
Wa!>b = ((+e)1.5 "10 7
The potential difference Va-Vb?
V
)(0.50m) = 0.75 "10 7 ev
m
= ? MeV
Va !Vb =
Wa!>b 7.5MeV
=
= 7.5MV
q
e
Finding the potential
Va=?
Vb=?
Vc=?
+12nC
-12nC
A dust (M=5.0µg, q=2.0nC) moves from
a (from rest) to b. What is the speed v at b?
Ka + Ua = Kb + Ub
1
0 + qVa = mv 2 + qVb
2
1 Q
V (r) =
4 πε 0 r
€
Q23.7
The electric potential due to a point charge
approaches zero as you move farther away from the
charge.
If the three point charges shown here lie at the
vertices of an equilateral triangle, the electric
potential at the center of the triangle is
Charge #2
+q
Charge #1
+q
y
–q
x
A.  positive.
B. negative.
C. zero.
D. not enough information given to decide
Charge #3
Q23.8
The electric potential due to a point charge
approaches zero as you move farther away from the
charge.
If the three point charges shown here lie at the
vertices of an equilateral triangle, the electric
potential at the center of the triangle is
Charge #2
–q
Charge #1
+q
y
–q
x
A.  positive.
B.  negative.
C. zero.
D. not enough information given to decide
Charge #3
Q23.9
Consider a point P in space where the electric potential is zero. Which statement is
correct?
A. A point charge placed at P would feel no electric force.
B. The electric field at points around P is directed toward P.
C. The electric field at points around P is directed away from P.
D. none of the above
E. not enough information given to decide
Example: electrical potential of a conducting sphere
A solid conducting sphere of radius R has a total charge q. What is V(r)?
Outside the sphere, is the E field
any different from a point charge
Located at the center?
For a point charge:
V =
U
1 q
=
q0 4 πε 0 r
For the sphere with r>R:
€ if r=R?
What
What if r<R?
What if it is an insulating sphere?
Example—oppositely charged parallel plates
•  What is V(y), as a function of y, between two oppositely charged
plates
•  If we chose Ub=0, then U(y)=q0Ey
•  V(y)=U(y)=Ey=?
•  What if Ub is not 0? What is Ua-Ub? Va-Vb?
Knowing E=σ/ε0, how
would you measure σ, the charge density
Example: Infinite Charged Conducting Cylinder
Just like an infinite line of charge,
1 λ
2πε 0 r
E r (r > R) =
The field is known, therefore,
b


b
E • dl = ∫ a E r dr
Va − Vb =
∫
Va − Vb =
λ
2πε 0
a
1
λ
rb
dr
=
ln
∫ a r 2πε r
0
a
b
Problem, if Vb=0 at infinity, then Vb is always infinite.
Why? How to solve it?
Define Vb=0 at r=R! (This is always arbitrary)
λ
Va − VR = Va =
2πε 0
∫
R
a
At r<=R, V?
At r>R, is it positive or negative
€
1
λ
R
dr =
ln
r
2πε 0 ra
Example: a ring of charge
We needed integral to derive the field previously.
E=
Qx
4 πε 0 ( x 2 + a 2 ) 3
Is it necessary to use integral to derive the potential?
V=
1
4 πε 0
∫
dq
r
r ≡ x 2 + a2
∴V =
1
4 πε 0 x 2 + a 2
∫ dq =
Q
4 πε 0 x 2 + a 2
What happens if you take the derivative of V: dV/dx=?
€
Potential Gradient
Knowing the V everywhere, we can measure the E everywhere too!
Va − Vb =
∫
b
a


E • dl = −
∫
b
a
dV


−dV = E • dl = (E x iˆ + E y ˆj + E z kˆ )(iˆ dx + ˆjdy + kˆdz)
−dV = E x dx + E y dy + E z dz What if the displacement is only along the x axis?
dy=dz=0!
This is how we define partial derivative!
⇒
∂V
∂x
∂V
Ey = −
∂y
∂V
Ez = −
∂z
Ex = −

∂V ˆ ∂ V ˆ ∂V
E = −(iˆ
+j
+k
)
∂x
∂y
∂z

∂ ˆ∂ ˆ∂
ˆ
∇
=
−(
i
+ j +k )
Operator grad/del
∂x
∂y
∂z


⇒ E = −∇V
The electric field is the negative of the gradient of V!
Can you go back to the ring of charge and derive E from V?
€
€
Equipotential surfaces and field lines
•  Contour lines on a topographic map:
equi-gravitational-potential-energy line (for a given mass)
•  Equipotential surface (Electric field): V is the same everywhere
Cross sections of equipotential surfaces
Field lines are always perpendicular to the equipotential surfaces, why?
Equipotential surfaces and field lines
• Is the vertical plane an equipotential surface?
• Why?
• Is the electric field the same on an equipotential
surface?
Examples for yes?
or no situations?
The surface and interior of a conductor
Why is the electric field right outside a conductor
surface always perpendicular to the surface?
How many methods can you use
to prove this?
Q23.10
Where an electric field line crosses an equipotential surface, the angle between the
field line and the equipotential is
A. zero.
B. between zero and 90°.
C. 90°.
D. not enough information given to decide
Q23.11
The direction of the electric potential gradient at a certain point
A. is the same as the direction of the electric field at that point.
B. is opposite to the direction of the electric field at that point.
C. is perpendicular to the direction of the electric field at that
point.
D. not enough information given to decide