Chapter 23: Electrostatic Energy & Capacitance Chapter 24: …shill/Teaching/2049 Fall11... ·...

Chapter 23: Electrostatic Energy & Capacitance Chapter 24: Electric Current & Ohm’s law

Tuesday September 27th

• Review of Mini Exam 2 • Review of Capacitance and Electrostatic Energy • Capacitors in series and parallel

• Demonstration and example • Dielectrics and capacitance

• Demonstration and explanation • Conductors under dynamic conditions

• Current, current density, drift velovity • Ohm’s law

Reading: up to page 410 in the text book (Ch. 24)

Capacitors • The transfer of charge from one terminal of the capacitor to the other creates the electric field.

• Where there is an electric field, there must be a potential difference, leading to the following definition of capacitance C:

C = Q

!V or Q = C!V

• Q represents the magnitude of the excess charge on either plate. Another way of thinking of it is the charge that was transferred between the plates.

SI unit of capacitance: 1 farad (F) = 1 coulomb/volt

(after Michael Faraday)

V (q)

q U

Energy stored in a Capacitor

U = q2

2C=

CV( )2

2C= 1

2 CV 2

= q2

2C= 1

2qC

q = 12 qV

Just like energy stored in a spring

Capacitors connected in series

1 2

1 1 1

eqC C C= +

1 1neq nC C

=∑In fact:

Capacitors connected in parallel

1 1q C V= Δ

2 2q C V= Δ1 2 eqq q C V+ = Δ

( )1 2 eqC C V C V+ Δ = Δ

1 2eqC C C= +

+q2 -q2

-q1 +q1

!V+(q1+q2) -(q1+q2)

!V

More Challenging Example

C1 C2

C3

C4 C5

A

B

What is capacitance between points A and B

A dielectric in an electric field

Electric dipoles in an electric field Non-polar atoms in an electric field

A dielectric in an electric field

!E =!E0 +

!E'

E = E0 - E' E' opposes E0 Linear materials: E' ! E, or E' = "eE

! E = E0 " #eE, or E0 = 1+ #e( )E

!e is the electric susceptibility (dimensionless)

A dielectric in an electric field

0 01 1 ( 1 )

1 e ee e

E E E κ χχ κ

⇒ = = = ++

Linear materials: E0 = 1+ !e( )E

!e is the electric susceptibility (dimensionless)" e is the dielectric constant (dimensionless)# =" e#o is the permittivity

A dielectric in an electric field

Linear materials: E0 = 1+ !e( )E

If E =

E0

! e

, then "V ="V0

! e

= 1! e

QdA#o

Isolated capacitor: Potential difference without dielectric

Actual potential difference with dielectric Potential decreases due to screening of field by surface charge

Capacitor with dielectric between plates

Linear materials: E0 = 1+ !e( )E

!V =

!V0

" e

= 1" e

QdA#o

Isolated capacitor:

! Ceff =

Q"V

=# e

$o Ad

=# eC

Capacitance increases:

Conductors in E-fields: dynamic conditions • If the E-field is maintained,

then the dynamics persist, i.e., charge continues to flow indefinitely.

• This is no longer strictly the domain of electrostatics. • Note the direction of flow of

the charge carriers (electrons).

Electrical current: I = dQ

dt! "Q

"tSI unit: 1 ampere (A) = 1 coulomb per second (C/s)

+ E

E

E

E E

I