Last%20time

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Last time Fields, forces, work, and potential

• Electric forces and work

Electric potential energy and electric potential
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Potential from electric field
VVo

• dV largest in direction of E-field.
• dV smallest (zero) perpendicular to E-field

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Equipotential lines

• Lines of constant potential
• In 3D, surfaces of constant potential

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Topographic map

• Each lines is constant elevation
• Same as constant gravitational potentialgh
  (energy mgh)
• Height interval between lines constant

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Electric field from potential

• Said before that

• Spell out the vectors

• This works for

Usually written
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Quick Quiz

• Suppose the electric potential is constant
  everywhere. What is the electric field?

1. Positive
2. Negative
3. Increasing
4. Decreasing
5. Zero

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Electric Potential - Uniform Field
B
Constant E-field corresponds to linearly
decreasing (in direction of E) potential
A
Here V depends only on x, not on y
x
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Check of basic cases

• Previous quick quiz uniform potential
  corresponds to zero electric field
• Linear potential corresponds to constant electric
  field

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Potential ( V ) of spherical conductor

• What is V of spherical conductor relative to
  infinity?
• Charge on surface ? spherical charge shell
• Gauss law ? E keQ / r2 in the radial direction
• V is work / Coulomb to bring point charge from 8

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Quick quiz
Previous result says conducting sphere of radius
R carrying charge Q is at a potential

• Two conducting spheres of diff radii connected by
  long conducting wire. What is approximately true
  of Q1, Q2?

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Connected spheres

• Since both must be at the same potential,

Smaller radius sphere has smaller charge
Surface charge densities?
Smaller sphere has larger surface charge density
Electric field?
Local E-field bigger at more sharply curved
(smaller R) regions
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Varying E-fields on conductor

• Larger electric fields near smaller radii
  surfaces.
• Large electric fields at sharp points,
• Strong fields can ionize air atoms.

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Potential and charge

• Have shown that a conductor has an electric
  potential, and that potential depends on its
  charge
• For a charged conducting sphere

Electric potential proportional to total charge




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Quick Quiz

• Consider this conducting object. When it has
  total charge Qo, its electric potential is Vo.
  When it has charge 2Qo, its electric potential

A. is Vo B. is 2Vo C. is 4Vo D. depends on shape
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Capacitance

• Electric potential of any conducting object
  proportional to its total charge.

• C capacitance
• Large capacitance need lots of charge to change
  potential
• Small capacitance small charge can change
  potential.

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Capacitors

• Where did the charge come from?
• Usually transferred from another conducting
  object, leaving opposite charge behind
• A capacitor consists of two conductors
• Conductors generically called plates
• Charge transferred between plates
• Plates carry equal and opposite charges
• Potential difference between platesproportional
  to charge transferred Q

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Definition of Capacitance

• Same as for single conductor
• but ?V potential difference between plates
• Q charge transferred between plates
• SI unit of capacitance is farad (F) 1 Coulomb /
  Volt
• This is a very large unit typically use
• mF 10-6 F, nF 10-9 F, pF 10-12 F

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How was charge transferred?

• Battery has fixed electric potential difference
  across its terminals
• Conducting plates connected to battery terminals
  by conducting wires.
• DVplates DVbattery across plates
• Electrons move
• from negative battery terminal to -Q plate
• from Q plate to positive battery terminal
• This charge motion requires work
• The battery supplies the work

DV
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Work done to charge a capacitor

• Requires work to transfer charge dq from one
  plate

• Total work sum of incremental work

• Work done stored as potential energy in capacitor

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Example Parallel plate capacitor

• Charge Q moved from right conductor to left
  conductor
• Charge only on inner surfaces
• Plate surfaces are charge sheets, each producing
  E-field

Uniform field between plates
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Quick Quiz

• Electric field between plates of infinite
  parallel-plate capacitor has a constant value
  ?/?o. What is the field outside of the plates?

1. ?/?o
2. ?/2?o
3. - ?/2?o
4. ?/4?o
5. 0

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What is potential difference?
Potential difference V-V- - (work to move
charge q        from plate to - plate) / q
d
-Q
Q
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What is the capacitance?
-Q
This is a geometrical factor
Q
Energy stored in parallel-plate capacitor
Energy density
d