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Physics 2112 Unit 7: Conductors and Capacitance. Today’s Concept: Conductors Capacitance. Comments. 3. Definition of Potential:. THERE ARE ONLY THREE THINGS YOU NEED TO KNOW TO DO ALL OF HOMEWORK. 1 . E = 0 within the material of a conductor:

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physics 2112 unit 7 conductors and capacitance

Physics 2112Unit 7: Conductors and Capacitance

Today’s Concept:

Conductors

Capacitance

comments
Comments

3. Definition of Potential:

THERE ARE ONLY THREE THINGS YOU NEED TO KNOW TO DO ALL OF HOMEWORK

1. E= 0 within the material of a conductor:

Charges move inside a conductor in order to cancel out the fields that would be there in the absence of the conductor. This principle determines the induced charge densities on the surfaces of conductors.

2. Gauss’ Law:

If charge distributions have sufficient symmetry (spherical, cylindrical, planar), then Gauss’ law can be used to determine the electric field everywhere.

CONCEPTS DETERMINE THE CALCULATION !

conductors
Conductors

The Main Points

  • Charges free to move
  • E=0 in a conductor
  • Surface = Equipotential
  • E at surface perpendicular to surface
checkpoint two spherical conductors 1
CheckPoint: Two Spherical Conductors 1

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.

VA > VB

VA = VB

VA < VB

checkpoint two spherical conductors 2
CheckPoint: Two Spherical Conductors 2

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.

The two conductors are now connected by a wire. How do the potentials at the conductor surfaces compare now?

VA > VB

VA = VB

VA < VB

checkpoint two spherical conductors 3
CheckPoint: Two Spherical Conductors 3

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.

What happens to the charge on conductor A after it is connected to conductor B by the wire?

QAincreases

QAdecreases

QAdoesnotchange

capacitor
Capacitor

Electric Circuit Element

  • Same uses as spring in mechanical system
  • smooth out rough spots
  • store energy
  • cause controlled oscillations
capacitor ii
Capacitor (II)

E

d

Simplest Example: Parallel plate capacitor

Define capacitance, C, such that:

-Q

+Q

How easy is it to stuff charge on the plates

How easy is it to stuff charge on the plates

How difficult is it to change the length of the spring

Q

Units of Farad, F = Coulomb/Volt

V

key points
Key Points
  • +Q and –Q always have same magnitude
  • Charges don’t more directly from one plate to the other
  • Charged from the outside
review of capacitance example
Review of Capacitance Example

y

x

First determine E field produced by charged conductors:

+Q

What is s ?

d

E

-Q

A = area of plate

Second, integrate E to find the potential difference V

As promised, V is proportional to Q !

  • Method good for all cases
  • Formula good for parallel plate only
example 7 1 capacitor
Example 7.1 (Capacitor)

A flat plate capacitor has a capacitance of C = 10pF and an area of A=1cm2. What is the distance between the plates?

checkpoint results charged parallel plates 1
CheckPoint Results: Charged Parallel Plates 1

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.

  • Compare Q1 and Q0.
    • Q1 < Q0
    • Q1 = Q0
    • Q1 > Q0
checkpoint results charged parallel plates 11
CheckPoint Results: Charged Parallel Plates 1

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.

  • Compare the capacitance of the two configurations in the above problem.
    • C1 > C0
    • C1 = C0
    • C1 < C0
example 7 2 linear capacitor
Example 7.2 (Linear Capacitor)

A capacitor is constructed from two conducting cylindrical shells of radii a1, a2, a3, and a4and length L(L>>ai).

What is the capacitance C of this capacitor ?

cross-section

a4

a3

a2

a1

metal

  • Conceptual Idea:

metal

Find V in terms of some general Q and divide Q out.

  • Plan:
    • Put +Qon outer shell and-Qon inner shell
    • Cylindrical symmetry: Use Gauss’ Law to calculate E everywhere
    • Integrate E to get V
    • Take ratio Q/V(should get expression only using geometric parameters (ai, L))
  • Limiting Case:
    • L gets bigger, C gets bigger
    • a2 –> a3, C gets bigger
example 7 2 linear capacitor1
Example 7.2 (Linear Capacitor)

A capacitor is constructed from two conducting cylindrical shells of radii a1, a2, a3, and a4and length L(L>>ai).

What is the capacitance C of this capacitor ?

cross-section

a4

a3

a2

a1

metal

metal

  • Do Limiting Cases Work?
    • L gets bigger, C gets bigger
    • a2 –> a3, C gets bigger
energy in capacitors
Energy in Capacitors

DU is equal to the amount of work took to put all the charge on the two plates:

example 7 3 energy in capacitor
Example 7.3 (Energy in Capacitor)

A 8uF parallel plate capacitor is has a potential different of 120V between its two sides. The distance between the plates is d=1mm.

What is the potential stored in the capacitor?

What is the energy density of the capacitor?