Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 11

Physics 212Physics 212Lecture 7Lecture 7

Today's Concept:Today's Concept:

Conductors and CapacitanceConductors and Capacitance

How are charges distributed on conductors? How are charges distributed on conductors? What is capacitance and how can we calculate it?What is capacitance and how can we calculate it?

Main Point 1

First, charges in a conductor will always move so as to create a zero electric field at all points in the conductor. Consequently, all conductors are equipotentials and the electric field at the surface of a conductor is always perpendicular to that surface.

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 22

Main Point 2

Second, the capacitance C of a system composed of two spatially separated conductors (called a capacitor) is defined in terms of the potential difference V between the conductors that is produced when each conductor carries an equal amount Q of oppositely signed excess charge. In particular, C is defined to be equal to Q/V.

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 33

Main Point 3

Third, energy is stored in electric fields. The energy density (i.e., the energy per unit volume) present in an electric field E is proportional to the square of the electric field (i.e., u = ½ e0E2). When a capacitor is charged, it has an energy U stored in the electric field between the conductors which is proportional to the product of the potential difference between the conductors and the separated charge (i.e., U = ½ QV)

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 44

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 55

Checkpoint 1aCheckpoint 1a

66

Two spherical conductors are separated by a large distance. They each carry the same positive charge Q. Conductor A has a larger radius than conductor B.

Compare the potential at the surface of conductor A with the potential at the surface of conductor B.A. VA > VB B. VA = VB C. VA < VB

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 66

Checkpoint 1bCheckpoint 1b

77

The two conductors are now connected by a wire. How do the potentials at the conductor surfaces compare now?A. VA > VB B. VA = VB C. VA < VB

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 77

Checkpoint 1cCheckpoint 1c

88

What happens to the charge on conductor A after it is connected to conductor B by the wire?A. QA increases B. QA decreases C. QA doesn’t change

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 88

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 99

Checkpoint 2aCheckpoint 2a

A) Q1 < QoB) Q1 = Qo C) Q1 > Qo

Two parallel plates of equal area carry equal and opposite charge Q0. The potential difference between the two plates is measured to be V0. An uncharged conducting plate (the green thing in the picture below) is slipped into the space between the plates without touching either one. The charge on the plates is adjusted to a new value Q1 such that the potential difference between the plates remains the same.

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1010

Checkpoint 2bCheckpoint 2b

A) CA) C11 > C > Coo B) CB) C11 = C = Coo C) CC) C11 < C < Coo

Two parallel plates of equal area carry equal and opposite charge Q0. The potential difference between the two plates is measured to be V0. An uncharged conducting plate (the green thing in the picture below) is slipped into the space between the plates without touching either one. The charge on the plates is adjusted to a new value Q1 such that the potential difference between the plates remains the same. What happens to C1 relative to C0?

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1111Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1111

CapacitanceCapacitanceCapacitance is defined for any pair of spatially separated Capacitance is defined for any pair of spatially separated conductorsconductors

99

QC

V

How do we understand this definition ???How do we understand this definition ???

+Q+Q

-Q-Q

dd

• Consider two conductors, one with excess charge = +Q Consider two conductors, one with excess charge = +Q and the other with excess charge = -Qand the other with excess charge = -Q

• These charges create an electric field in the space between These charges create an electric field in the space between themthem

EE

• This potential difference should be proportional to Q !!This potential difference should be proportional to Q !!• The ratio of Q to the potential difference is the capacitance The ratio of Q to the potential difference is the capacitance and only depends on the geometry of the conductorsand only depends on the geometry of the conductors

VV

• We can integrate the electric field between them to find the We can integrate the electric field between them to find the potential difference between the conductorpotential difference between the conductor

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1212

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1313

Example ProblemExample ProblemTwo parallel plates of area Two parallel plates of area AA separated by a distance separated by a distance dd carry equal an carry equal an opposite charge opposite charge QQ00. An uncharged conducting plate having thickness . An uncharged conducting plate having thickness tt is slipped midway between the plates.is slipped midway between the plates.

+Q+Q00

-Q-Q00

dd ttdd

+Q+Q11

-Q-Q11

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1414

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1515Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1515

BANGBANG

3131

Energy in CapacitorsEnergy in Capacitors

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1616

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1717

CalculationCalculation

A capacitor is constructed from two conducting cylindrical shells of radii a1, a2, a3, and a4 and length L (L >> ai).

What is the capacitance C of this capacitor ?

metal

metal

a1

a2

a3

a4

• Conceptual Analysis:

•Strategic Analysis

cross-sectioncross-section

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1818

Physics 212 Lecture 7, Slide Physics 212 Lecture 7, Slide 1919