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 –3 Q , 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 ʼ 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 ʼ s law would be useful for finding the electric field in all of these cases.

•   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
= #
!
E • d
!
A =
$ q
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

•   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

•   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)

•   The work done raising a basketball against gravity depends only on the potential energy, how high the ball goes.
(the definition of a conservative force)
W a !> b
= " b
!
F • d l
!
= !# U = !
( U b a
•   A point charge moving in a field exhibits similar behavior (
).
!
U a
)
Only the displacement in y direction matters!
U = qEy d = y a
!
y b
W a !> b
= qEd = !" U = !
( U b
!
U a
) = " K
How fast will the charge move at b?

Positve 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!
Negative Charge

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.
+ q
!
E
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.
Motion
!
E
+ 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.
– q
!
E
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.
Motion
!
E
– 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
F =
4 πε
0 qq
0 r 2
W a − > b
=
a b
1
4 πε
0 qq
0 r 2 dr = qq
0
4 πε
0
(
1 r a
−
1
) r b
If we define U=0 at r  infinity
1
U =
4 πε
0 qq
0 r
€
What if the path from a  is not a straight line?

•   The path does not matter!
•   Here is the proof!
W a !> b
=
W a !> b
=
" a b
!
F • d l
!
= " a b
F cos !
dl
" a b
1
4 "#
0 cos !
dl # dr!
qq
0 r 2 cos !
dl
U =
1
4 !"
0 qq
0 r
It DEPENDS on the signs!

•   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=3.00x10
6 m/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
1
2 mv b
2
=
= U a
" U b
=
1
2 mv
2 a
+
( + e )( + 2 e )
4 !"
( + e )( + 2 e )
4 !"
0
0
(
1 r a
(
1 r a
"
1
) r b
"
1
) r b
How far can an electron go?

•   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
U =
4 q
0
!"
0
!
i q r i i
What about the PE between all of them: what’s needed to
Assemble these charges together?
U =
1
4 !"
0
!
i < j q i q j r ij
Two charges : q
1
=+e at x=a.
=-e at x=0, and
q
2
How much work is needed to bring q
3
=+e from infinity to x=2a?
What’s the total potential energy of the three charge system?

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
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
Charge #1
+ q y x
A. positive.
B. negative.
C. zero. D. not enough information given to decide
Charge #2
+ q
– q
Charge #3

Q23.6
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
Charge #1
+ q y x
A. positive.
B. negative.
C. zero. D. not enough information given to decide
Charge #2
– q
– q
Charge #3

€
The electrical potential: What does 1.5Volt battery mean?
•   Electric potential: Electric potential energy per unit charge
U
V = q
0
V = volt = J / C = joule / coulomb
V ab
= V a
− V b
= ( U a
− U b
) / q
0
•   V ab
: 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…)

Point Charge
V =
U q
0
Multiple Charge:
1
=
4 πε
0 q r
V =
U q
0
=
1
4 πε
0
∑ i q r i
Is this a vector sum or scalar sum?
Continuous Charges:
V =
U q
0
1
=
4 πε
0
∫ dq r
=
1
4 πε
0
∫
If the electric field is known:
V a
− V b
= ∫ a b

E • d
 l = ∫ a b
E cos φ dl
ρ (

) d
 r
€
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
-19 C) is accelerated from point a to be (distance d=0.50m). E=1.5x10
7 V/m=1.5x10
7 N/C (uniform)
What is the work done by the field?
W a !
> b
= Fd = ( qE ) d
W a !
> b
= ((
The potential difference V a
+ e )1.5
-V b
?
" 10
= ?
MeV
7
V m
)(0.50
m ) = 0.75
" 10 7 ev
V a
!
V b
=
W a !
> b q
=
7.5
MeV e
= 7.5
MV

V a
=?
V b
V c
=?
=?
+12nC
-12nC
A dust (M=5.0
µ g, q=2.0nC) moves from a (from rest) to b. What is the speed v at b?
K a
+ U a
0 + qV
V ( r ) = a
= K b
+ U b
1
=
1
2 mv 2
Q
+ qV b
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 y
Charge #1
+ q x
A.
positive.
B. negative.
C. zero.
D. not enough information given to decide
Charge #2
+ q
– q
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 y
Charge #1
+ q x
A.
positive.
B.
negative.
C. zero.
D. not enough information given to decide
Charge #2
– q
– q
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

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:
U
V = q
0
1
=
4 πε
0 q r
For the sphere with r>R:
What if r<R?
What if it is an insulating sphere?

•   What is V(y), as a function of y, between two oppositely charged plates
•   If we chose U b
=0, then U(y)=q
0
Ey
•   V(y)=U(y)=Ey=?
•   What if U b
is not 0? What is U a
-U b?
V a
-V b
?
Knowing E= σ / ε
0
, how would you measure σ , the charge density

€
Example: Infinite Charged Conducting Cylinder
Just like an infinite line of charge,
E r
( r > R ) =
1
2 πε
0
λ r
The field is known, therefore,
V a
− V b
= ∫ a b

E • d
 l = ∫ a b
E r dr
V a
− V b
λ
=
2 πε
0
∫ a b
1 r dr =
λ
2 πε
0 ln r b r a
Problem, if V b
=0 at infinity, then V b
is always infinite.
Why? How to solve it?
Define V b
=0 at r=R! (This is always arbitrary)
V a
− V
R
= V a
=
2
λ
πε
0
∫ a
R
1 r dr =
2
λ
πε
0 ln r a
R
At r<=R, V?
At r>R, is it positive or negative

We needed integral to derive the field previously.
Qx
E =
4 πε
0
( x 2
+ a 2 ) 3
Is it necessary to use integral to derive the potential?
1
V = dq
4 πε
0
∫ r r ≡ x 2
+ a 2
∴ V =
4 πε
0
1 x 2
+ a 2
∫ dq =
4 πε
0
Q x 2
+ a 2
What happens if you take the derivative of V: dV/dx=?
€

€
Knowing the V everywhere, we can measure the E everywhere too!
V a
− V b
=
a b

E • d
 l =
b
− dV a
− dV =

E • d
 l = ( E x
ˆ
+ E y
ˆ
+ E z
ˆ i dx +
ˆ
+
ˆ
)
− 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!
⇒
E
E
E x y z
= −
∂ V
∂ x
= −
∂ V
∂ y
= −
∂ V
∂ z
Operator grad/del

E = − i
∂ V
∂ x

∇ = − i
∂
∂ x
⇒
+

E = −

∇ V
+
∂
∂
∂
∂ y
V y
+
+
∂
∂ z
∂
∂
)
V z
)
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?
€

•   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?

•   Is the vertical plane an equipotential surface?
•   Why?
•   Is the electric field the same on an equipotential
surface?
Examples for yes? or no situations?

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