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Announcement 1

You MUST show up at the correct exam room:

Aaa-Hal ARC-103

Haq-Mou Hill-114

Mug-Seh PHY-LH

Sen-Zzz SEC-111

Physics 227, Common Hour Exam #1

Thursday, October 5, 2017

9:50 to 11:10 PM (at night)

on Busch campus

Announcement 2

Please urgently e-mail your requests to Prof. Montalvo

( rm1255@physics.rutgers.edu ):

- for extra time on the exam because of disability

(you need a letter from Dean documenting disability);

- for a makeup exam, if you have a legitimate reason

(makeup will be given on Wed. next week – double check

with Prof. Montalvo).

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Announcement 3

The best way to prepare for the exam:

- do practice exams from prior years (they are posted on

the course website);

- go through i-clicker and quizes;

- finish up all homework and/or collaborative assignments

that are overdue;

- come to office hours (prepared with your list of questions

ready!)

Lecture 7. Capacitors and Electric Field Energy

10

At metallic surfaces the electric field lines

are perpendicular to the surface.

Metal

���� ?

Metal

���� would drive

surface current

Metal

���� the only option

What can we conclude about the electric

potential distribution over the surface?

Conducting Surface ≡≡≡≡ Equipotential Surface

The surface of a conductor is always an equipotential surface (in electrostatics!):

when we move a test charge along the surface, no work is done (�� � ���).

Conducting Surface = Equipotential Surface

11

Since E=0 inside, any point in the volume of a conductor is

at the same potential as its surface.

E=0

Capacitors

+Q

Capacitor: a system of two conducting surfaces

(electrodes), the net charge is zero.

Important: because the surfaces are conducting,

they are equipotential, and

electrodes

-Q

12

V1 V2

�

Parallel-plate

capacitor:

∆� � � � !"!#$%&'!

(!"!#$%&'!∙ ���

- is the same for any

path between the

electrodes.

13

Capacitance

Capacitance: * ≡ ,∆� �

,- � !"!#$%&'!(!"!#$%&'! ∙ ���

Again, this definition makes sense for conducting surfaces only.

13

+Q

electrodes

-Q

V1V2

�

Units of capacitance are Farads, F:

[C]/[V] = [F]=

Analogy with water in a test tube:

,– total volume of water in the test tube

C – cross-section area (capacitance) of the test tube

∆� – level (height) of water in the test tube

Capacitance: , ≡ *⋅∆�

, ≡ *⋅∆�

Capacitance of a parallel-plate capacitor

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* ≡ ,∆� �

,- � !"!#$%&'!(!"!#$%&'! ∙ ���

� � ./0 �

,/01

2

3 � �2

4� �

�

d

∆� � �� � ,�/01

* � /01�

A ~ 1 m2

d ~ 10 µm = 10-5m

* 5 9 ∙ 10 9: �;1;:

1 ∙ 10 <; 5 1 ∙ 10 =�

/0 5 9 ∙ 10 9: �;

Recipe #1 for computing C : for a given Q, calculate

∆V, and use the definition of capacitance.

Typical numbers:

, ≡ *⋅∆�

Capacitance of a coax cable (cylindrical capacitor)

17

-Q

>:

>9

+Q

∆� � � � !"!#$%&'!

(!"!#$%&'!∙ ��� � � ?

2AB0CDE

DF�C � ?

2AB0 �G>:>9

*HIJC�JGKLM�N � ,� �?�� �

2AB0�G >:>9

�

Energy Stored in a Capacitor

18

How much work it takes

to charge a capacitor?

O � � PQ ∙ Q/01 ∙ �RST

RS0� �/01 ∙

,:2 � ,

:2*

The potential energy stored

in the capacitor:

Initial

state

Final

state

-e,U � 0

� � 0

,V � ,

� � ,/01

PO � PQ ∙ � Q ∙ � � PQ ∙ Q/01 ∙ � W 0force

X � ,:2* �

*�:2 � ,�2

The capacitors are discharged through a coil which

produces the magnetic field ~ 100 T for a few milliseconds.

50 MJ world's largest capacitor bank

(Dresden High Magnetic Field Lab)

� �

,

Energy Stored in Electric Field

23

Z[ � *�:2 ∙ 11� �

/01�

�� :2

11� �

12 /0�:

- this result is general,

applies also to time-

dependent electric

fields.

� � ,/01

C

Energy density (U/volume):

Z[ � 12 /0�:

Total energy of the

electric field:X[ � � 1

2 /0�: C �\]""^_]#!

In electrostatics, both approaches are equivalent. In

electrodynamics, the alternating e.-m. field can exist with no

obvious relevance to charges, so it’s preferable to think that

the energy is stored in the electric field.

energy stored

in the field

energy stored

in charges?

Field Energy vs. Energy of Interactions btw Charges

26

UE takes into account the work on assembling the individual charges, whereas

Uint doesn’t include the self-energies of these charges (doesn’t include the

work on assembling the individual charges).

- the field energy is always positive (∝ E2).

The potential energy of

interaction Uint between

charges can be either

positive or negative.

X[ � � 12 /0�: C �\]""^_]#!

�

Example: the electric field energy of a charged metal sphere.

� 12 /0

,4A/0C:

:4AC:�C

b

D� c ,

:2>

C

X[ W 0Potential energy of

interaction between

the charged electrodes

- sign?

Example

27

1. Charged spheres are far away (XUd$ � 0).

X$&$ ∞ � � 12 /0�: C �\]""^_]#!

� c ,:2> 4 c

,:2> � c

,:>

4,,

X$&$ � X9 4 X: 4 XUd$

2. Charged spheres are nearby.

XUd$ � c,:C

Two dielectric spheres with uniform surface charge density.

X$&$ ∞ W X$&$ C W 0

Conclusion

28

Next time: Lecture 8. Capacitors and Dielectrics

§§ 24.2, 24.4, 24.5

Conductors' surfaces are equipotential surfaces.

Capacitors.

Energy stored by a capacitor.

Electric field energy.

Appendix I. Self-Capacitance

29

One can consider the capacitance of an isolated conductor (assuming that the second

electrode is infinitely far away), self-capacitance. For example, for a metal sphere:

∆� � � �b

(!"!#$%&'!∙ ��� � ,

4AB0> * F � ,∆� � 4AB0> 5 10 90> m

Capacitance of the Earth (R ~ 6,400 km): *[]%$h 5 10 iFCapacitance of the top electrode of a van de Graaff generator (R ~ 0.2 m): 20 pF.

Appendix II. Faraday Cage vs. Grounding

+Q

-

-

-

E=0

Faraday cage:

total charge = 0,

field inside = 0,

equipotential surface

at V=???The Earth:

a huge conductor at a

(reasonably) constant

potential

(often considered as a

reference V = 0).

V = 0

wire

e-

E = 0What is the total charge

of the grounding shell?

Appendix III. Method of Mirror Images

31

V=0V

V=-1VV=1V

The equation for the potential has a unique solution if the boundary

conditions are fixed. These conditions are the same for the left half-

spaces of the figures. Thus, the electric field (to the right of the metal

surface) is also the same!

V=0V

Force of attraction between a point charge and

a grounded conducting surface:

� � 14AB0

,:2� :

2�