From last time(s)

1
From last time(s)

• Electric charges, forces, and fields
• Motion of charged particles in fields.

Today

• Work, energy, and (electric) potential
• Electric potential and charge
• Electric potential and electric field.

No honors lecture this week
2
Forces, work, and energy

• Particle of mass m at rest
• Apply force to particle - what happens?
• Particle accelerates

• Stop pushing - what happens?
• Particle moves at constant speed
• Particle has kinetic energy

3
Work and energy

• Work-energy theorem
• Change in kinetic energy of isolated particle
  work done

• Total work

4
Electric forces, work, and energy

• Consider bringing two positive charges together
• They repel each other
• Pushing them together requires work
• Stop after some distance
• How much work was done?

5
Calculating the work

• E.g. Keep Q2 fixed, push Q1 at constant velocity
• Net force on Q1 ?
• Force from hand on Q1 ?

Zero
Q1
R
Q2
xfinal
xinitial

• Total work done by hand

6
Conservation of Energy

• Work done by hand

for pos charges
Where did this energy go?
Energy is stored in the electric field as
electric potential energy
7
Electric potential energy of two charges

• Define electric potential energy U so that

Works for a two-charge system if

• Define potential energy at infinite separation
  0

for two charges
Then
Units of Joules
8
Quick Quiz

• Two balls of equal mass and equal charge are held
  fixed a distance R apart, then suddenly released.
  They fly away from each other, each ending up
  moving at some constant speed.
• If the initial distance between them is reduced
  by a factor of four, their final speeds are

1. Two times bigger
2. Four times bigger
3. Two times smaller
4. Four times smaller
5. None of the above

9
More About U of 2 Charges

• Like charges ? U gt 0 and work must be done to
  bring the charges together since they repel (Wgt0)
• Unlike charges ? U lt 0 and work is done to keep
  the charges apart since the attract one the other
  (Wlt0)

10
Electric Potential Energy of single charge

• Work done to move single charge near charge
  distribution.
• Other charges provide the force, q is charge of
  interest.

q

q1

q2


q3
Superposition of individual interactionsGeneraliz
e to continuous charge distribution.
11
Electric potential
Electric potential ?U energy proportional to
charge q
Electric potential

• Electric potential V usually created by some
  charge distribution.
• V used to determine electric potential energy U
  of some other charge q
• V has units of Joules / Coulomb Volts

12
Electric potential of point charge

• Consider one charge as creating electric
  potential, the other charge as experiencing it

q
Q
13
Electric Potential of point charge

• Potential from a point charge
• Every point in space has a numerical value for
  the electric potential

y
Q
x
14
Potential energy, forces, work

• UqoV
• Point B has greater potential energy than point A
• Means that work must be done to move the test
  charge qo from A to B.
• This is exactly the work to overcome the Coulomb
  repulsive force.

Work done qoVB-qoVA
Differential form
15
Quick Quiz

• Two points in space A and B have electric
  potential VA20 volts and VB100 volts. How much
  work does it take to move a 100µC charge from A
  to B?

1. 2 mJ
2. -20 mJ
3. 8 mJ
4. 100 mJ
5. -100 mJ

16
V(r) from multiple charges

• Work done to move single charge near charge
  distribution.
• Other charges provide the force, q is charge of
  interest.

q1
q2
q
q3
Superposition of individual electric potentials
17
Quick Quiz 1

• At what point is the electric potential zero for
  this electric dipole?

A
B

1. A
2. B
3. Both A and B
4. Neither of them

18
Superposition the dipole electric potential

• Superposition of
• potential from Q
• potential from -Q

V in plane
19
Electric Potential and Field for a Continuous
Charge Distribution

• If symmetries do not allow an immediate
  application of the Gauss law to determine E
  often it is better to start from V!
• Consider a small charge element dq
• The potential at some point due to this charge
  element is
• To find the total potential, need to integrate
  over all the elements
• This value for V uses the reference of V 0 when
  P is infinitely far away from the charge
  distribution

20
Quick Quiz

• Two points in space have electric potential
  VA200V VB150V. A particle of mass 0.01kg and
  charge 10-4C starts at point A with zero speed. A
  short time later it is at point B.
• How fast is it moving?

1. 0.5 m/s
2. 5 m/s
3. 10 m/s
4. 1 m/s
5. 0.1 m/s

21
E-field and electric potential

• If E-field known, dont need to know about
  charges creating it.
• E-field gives force
• From force, find work to move charge q

q




Electric potential
22
Potential of spherical conductor

• Zero electric field in metal -gt metal has
  constant potential
• Charge resides on surface, so this is like the
  spherical charge shell.
• Found E keQ / R2 in the radial direction.
• What is the electric potential of the conductor?

Integral along some path, from point on surface
to inf.
23
Electric potential of sphere
So conducting sphere of radius R carrying charge
Q is at a potential

• Conducting spheres connected by conducting wire.
• Same potential everywhere.

Q1
Q2
R1
R2
But ?not same everywhere
24
Connected spheres

• Since both must be at the same potential,

Charge proportional to radius
Surface charge densities?
Surface charge density proportional to 1/R
Electric field? Since
Local E-field proportional to 1/R (1/radius of
curvature)
25
Varying E-fields on conductor

• Expect larger electric fields near the small end.
  Can predict electric field proportional to local
  radius of curvature.
• Large electric fields at sharp points, just like
  square
• Fields can be so strong that air is ionized and
  ions accelerated.

26
Quick Quiz

• Four electrons are added to a long wire. Which of
  the following will be the charge distribution?

27
Conductors other geometries

• Rectangular conductor (40 electrons)
• Edges are four lines
• Charge concentrates at corners
• Equipotential lines closest together at corners
• So potential changes faster near corners.
• So electric field is larger at corners.

28
E-field and potential energy
29

• What is electric potential energy of isolated
  charge?

Zero
30
The Electric Field

• is the Electric Field
• It is independent of the test charge, just like
  the electric potential
• It is a vector, with a magnitude and direction,
• When potential arises from other charges,
  Coulomb force per unit charge on a test charge
  due to interaction with the other charges.

Well see later that E-fields in electromagnetic
waves exist w/o charges!
31
Electric field and potential
Said before that

• Electric field strength/direction shows how the
  potential changes in different directions
• For example,
• Potential decreases in direction of local E field
  at rate
• Potential increases in direction opposite to
  local E-field at rate
• potential constant in direction perpendicular to
  local E-field

32
Potential from electric field

• Electric field can be used to find changes in
  potential
• Potential changes largest in direction of
  E-field.
• Smallest (zero) perpendicular to E-field

VVo
33
Quick Quiz 3

• Suppose the electric potential is constant
  everywhere. What is the electric field?

1. Positive
2. Negative
3. Zero

34
Electric Potential - Uniform Field

E cnst

• Constant E-field corresponds to linearly
  increasing electric potential
• The particle gains kinetic energy equal to the
  potential energy lost by the charge-field system

35
Electric field from potential

• Said before that
• Spell out the vectors
• This works for

Usually written
36
Equipotential lines

• Lines of constant potential
• In 3D, surfaces of constant potential

37
Electric Field and equipotential lines for and
- point charges

• The E lines are directed away from the source
  charge
• A positive test charge would be repelled away
  from the positive source charge

The E lines are directed toward the source
charge A positive test charge would be attracted
toward the negative source charge
Blue dashed lines are equipotential
38
Quick Quiz 1
Question How much work would it take YOU to
assemble 3 negative charges?

1. W 19.8 mJ
2. W 0 mJ
3. W -19.8 mJ

-3mC
5 m
5 m
Likes repel, so YOU will still do positive work!
-1mC
-2mC
5 m
39
Work done to assemble 3 charges
Similarly if they are all positive

• W1 0

• W2 k q1 q2 /r

3.6 mJ
(9?109)(1?10-6)(2?10-6)/5

• W3 k q1 q3/r k q2 q3/r
• (9?109)(1?10-6)(3?10-6)/5 (9?109)(2?10-6)(3?1
  0-6)/5 16.2 mJ
• W 19.8 mJ
• WE -19.8 mJ
• UE 19.8 mJ

q3
3C
5 m
5 m
2C
1C
q2
5 m
q1
40
Quick Quiz 2
The total work required for YOU to assemble the
set of charges as shown below is

1. positive
2. zero
3. negative

41
Why ?U/qo ?

• Why is this a good thing?
• ?V?U/qo is independent of the test charge qo
• Only depends on the other charges. ?V arises
  directly from these other charges, as described
  last time.
• Last weeks example electric dipole potential

• Superposition of
• potential from Q
• potential from -Q

42
Dipole electric fields

• Since most things are neutral, charge separation
  leads naturally to dipoles.
• Can superpose electric fields from charges just
  as with potential
• But E-field is a vector, -add vector components

43
Quick Quiz 2
In this electric dipole, what is the direction of
the electric field at point A? A) Up B) Left
C) Right D) Zero
A
44
Dipole electric fields
Note properties of E-field lines

45
Conservative forces
Fg

• Conservative Forces the work done by the force
  is independent on the path and depends only on
  the starting and ending locations.
• It is possible to define the potential energy U
• Wconservative -D U Uinitial - Ufinal
  -(Kfinal - Kinitial) -DK

46
Potential Energy of 2 charges

• Consider 2 positive charged particles. The
  electric force between them is
• The work that an external agent should do to
  bring q2 at a distance rf from q1 starting from a
  very far away distance is equal and opposite to
  the work done by the electric force.
  Charges repel
  ?Wgt0!

F
r12
47
Potential Energy of 2 charges

• Since the 2 charges repel, the force on q2 due to
  q1
• F12 is opposite to the direction of motion
• The external agent F -F12 must do positive
  work!
• W gt 0 and the work of the electric force WE lt 0

F
dr
r12
48
Potential Energy of 2 charges

• Since WE -DU Uinitial - Ufinal -W ? W
  DU
• We set Uinitial U(?) 0 since at infinite
  distance the force becomes null
• The potential energy of the system is

49
More than two charges?
50
U with Multiple Charges

• If there are more than two charges, then find U
  for each pair of charges and add them
• For three charges