1

§16.5 Motion of a Point Charge in a Uniform E-Field

Q) What is E-field around a metal plate w/ +Q?

+

+

+

+

Q) A metal plate w/ –Q?

Fig. 16.34

Parallel metal plates uniform E

Charge +q & mass m

“Cathode Ray Tube” (TV)

“Electron gun”

4

Charge +q & mass m.

E

Use kinematic equations w/ constant a from Ch. 4:

€

Δv = aΔt

Δx = 12 (v f − v i)Δt

Δx = v iΔt +12 aΔt

2

v f2 − v i

2 = 2aΔx

€

a =q

mE

5

Example: What electric field is needed to keep an electron suspended in the air against gravity? (a) Direction?(b) Strength?(c) Would a proton require the same field?

Example (PP 16.48): An electron is placed in a uniform electric field of strength 232 N/C. If the electron is at rest at the origin of a coordinate system at t = 0 and the electric field is in the positive direction, what are the x- and y-coords of the electron at t = 2.3 ns? The velocity?

6

§16.6 Conductors in Electrostatic Equilibrium

• Conductors are easily polarized: free electrons move around freely inside the material.

• Any charges placed on a conductor will arrange themselves in a stable, unmoving distribution: electrostatic equilibrium.

• For a conductor in electrostatic equilibrium:1) The E-field inside it is zero (no field lines)2) Any net charge must reside on the surface3) Just outside the surface, E is perpendicular to the surface4) Any excess charge will accumulate where the surface is highly curved (i.e. a sharp point): E is strongest there.

Put 16 nC on the following surface:Q) Where will charges go?

Q) What will the E-field look like?

7

Lightning rod

8

Chapter 18: Electric Potential• Electric Potential Energy

• Electric Potential (Voltage)

• How are the E-field and Electric Potential related?

• Motion of Point Charges in an E-field

• Capacitors

• Dielectrics

More help: SPS drop inMW 8:30-9:30amTR 11am-noon178 Overman Hall

Canvas goodiesCanvas goodies

9

§17.1 Electric Potential EnergyElectric potential energy (PEe) is:

• energy stored in the electric field,

• work (W=F.d) done to put charges in place.

+q

€

PEe =kq1q2

r

m

€

PEg = mgh

h

+Q +Q

-q

Example: Two point charges, Q = +6.0 C and q = +5.0 C are separated by 15.0 m.

• What is their potential energy?

• If Q is fixed and q is free to move, what will q do?

• How does q’s motion affect the potential energy? Explain in terms of conservation of energy.

10

11

Q) What is the potential energy of three point charges arranged as a right triangle?

12r

2q

1q 3q13r

23r

Q) What if there are four charges?

€

PE tot = PE i∑ = PE12 + PE13 + PE23 + ...

(scalar sum)

€

PE12 =kq1q2

r12

€

PE13 =kq1q3

r13

€

PE23 =kq2q3

r23

12

§17.2 Electric PotentialElectric potential is the electric potential energy per unit charge:

• scalar

• 1 V = 1 J/C.

For a point charge Q:

€

V =PEeqtest

=kQ

r

When a charge q moves through a potential difference of ΔV, its potential energy change is ΔPEe = qΔV.

€

V =PEeqtest

13

Q

b

a

c

e

d

g

f

Example: A charge Q = +1 nC is placed somewhere in space far from other charges. Take ra = rb = rc = rd = 1.0 m and re = rf = rg = 2.0 m.

(a) Compare the potential at points d and g.

(b) Compare the potential at points a and b.

14

Q

b

a

c

e

d

g

f

Example: A charge Q = +1 nC is placed somewhere in space far from other charges. Take ra = rb = rc = rd = 1.0 m and re = rf = rg = 2.0 m.

(c) Place a charge of +0.50 nC at point e. What will the change in potential (ΔV) be if this charge is moved to point a?

(d) What is the change in potential energy (ΔPE) of the +0.50 nC charge ?

15

§17.3 The Relationship between E and V

Q

b

a

c

e

d

g

f

+9 V

+4.5 V

Equipotentials: surfaces of equal potential.

16

E points in direction of maximum potential decrease.

E is perpendicular to the equipotential surfaces.

Q

b

a

c

e

d

g

f

+9 V +4.5 VE

Q) What is V at 3m? At 0.5 m?

Fig. 17.19

Q: What do the equipotentials look like around a – charge?

18

Equipotentials and field lines for a dipole:

19

Uniform E-field:

E

Equipotential surfaces

V1 V2 V3 V4

Edq

UV e

ΔΔ Where d is the distance

over which ΔV occurs.

20

Example: Two parallel plates are separated by 2.0 mm. One is at a potential of 3000.0 V while the other is at 0.0 V. What is the E-field between them?

Q) Why is E negative?

Hollow Conducting Sphere (radius = R):

(Similar for other hollow

shapes)

22

Van de Graaff generator

23

§17.4 Moving Charges

When only electric forces act on a charge, its total mechanical energy, E, will be conserved:

€

E i = E f

€

K i +U i =K f +U f

24

Example (PP 17.40): Point P is at a potential of 500.0 kV and point S is at a potential of 200.0 kV. The space between these points is evacuated. When a charge of +2e moves from P to S, by how much does its kinetic energy change?

(b) If the particle has a mass of 2.0x10-9 kg and starts from rest at P, what is its speed at S?

25

Example (text problem 17.41): An electron is accelerated from rest through a potential difference. If the electron reaches a speed of 7.26106 m/s, what is the potential difference?

26

Chapter 17: Electric Potential• Electric Potential Energy

• Electric Potential

• How are the E-field and Electric Potential related?

• Motion of Point Charges in an E-field

• Capacitors

• Dielectrics

For Mon recitation:• do Online Problems• do Practice Problems:

• Ch 17 (pp.634-7)42, 70, 83, 87, 91

Lab: 2.04 (E-field) this week• Read instructions• Do Pre-Lab & turn in• 2.05 (Current) next week

• Exam #1 (Ch 12, 16, 17)Wed Sep 12, 7:30-8:45pm, 095 Overman Hall

Free Tutoring & StudySee BlackBoard/C.I.

Practice Exam on BB

27

§17.5 CapacitorsA capacitor stores electric potential energy by storing separated (+) and (–) charges.

Work must be done to separate the charges.

Parallel plate capacitor

+ + + ++ + +

– ––––––

Why?

Fig. 17.22

30

VQ

VE

QE

ΔΔ

For a parallel plate capacitor:

+ + + ++ + +

– ––––––

ΔV

Or

Q = CΔV

where the proportionality constant C = capacitance

[ Farad = C/V ]

31

Note: C is a property of the device,

• it depends on A & d,

• “capacity” to hold charge.

What is the capacitance for a parallel plate capacitor?

€

ΔV = Ed = (4πkσ )d = 4πkQ

A

⎛

⎝ ⎜

⎞

⎠ ⎟d

∴Q =A

4πkd

⎛

⎝ ⎜

⎞

⎠ ⎟ΔV =CΔV

where C =A

4πkd.

32

Example (PP 17.56): A parallel plate capacitor has a capacitance of 1.20 nF. There is a charge of magnitude 0.800 C on each plate.

(a) What is the potential difference between the plates?

(b) If the plate separation is 0.3 mm, what is the area?

(c) If the plate separation is doubled while the charge is kept constant, what will happen to the potential difference, and to the potential energy stored in the capacitor?

33

§17.6 Dielectrics

I. Air-filled capacitor:

Increase Q increase E

Atoms in air b/w plates gets polarized:

Eventually electrons pulled off (ionized),

Charge arcs across gap = “breakdown”

Need a better insulator!

dielectric strength (kV/mm)

+ + + ++ + +

– ––––––

34

II. Add a dielectric w/

dielectric constant

Atoms polarize

Charge separation at ends

Reduces E inside dielectric

Can add more Q to plates

Higher C = Q/ΔV

+ + + ++ + +

– ––––––

€

C = κA

4πkd= κC0

36

Example (PP 17.71): A capacitor can be made from two sheets of aluminum foil separated by a sheet of waxed paper. If the sheets of aluminum are 0.3 m by 0.4 m and the waxed paper, of slightly larger dimensions, is of thickness 0.030 mm and has = 2.5, (a) what is the capacitance of this capacitor?

(b) How much charge can it hold before breakdown?

(c) How much energy is stored at this point?

McGuiver?!

37

§17.7 Energy Stored in a Capacitor

A capacitor will store energy equivalent to the amount of work that it takes to separate the charges.

38

(Sub in Q = CΔV)

C

Q

VC

VQU

2

2

12

1

2

2

The energy stored in the electric field between the plates is:

}Summary:

• C is set by the device (A, d, )

• ΔV is set by the strength of the battery (“pump”)

• Q and U depend on C and ΔV.

39

Example (PP 17.79): A parallel plate capacitor is composed of two square plates, 10.0 cm on a side, separated by an air gap of 0.75 mm.

(a) What is the charge on this capacitor when the potential difference is 150 volts?

(b) What energy is stored in this capacitor?

40

Summary

•Electric Potential Energy

•Electric Potential

•The Relationship Between E and V

•Motion of Point Charges (conservation of energy)

•Parallel Plate Capacitors (capacitance, dielectrics, energy storage)

41

§16.6 Conductors in Electrostatic Equilibrium

• Conductors are easily polarized: free electrons move around freely inside the material.

• Any charges placed on a conductor will arrange themselves in a stable, unmoving distribution: electrostatic equilibrium.

• For a conductor in electrostatic equilibrium:1) The E-field inside it is zero (no field lines)2) Any net charge must reside on the surface3) Just outside the surface, E is perpendicular to the surface4) Any excess charge will accumulate where the surface is highly curved (i.e. a sharp point): E is strongest there.

Put 16 nC on the following surface:Q) Where will charges go?

Q) What will the E-field look like?

42

Lightning rod