Capacitance

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Capacitance

• Alan Murray

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Some Capacitors
conductor
insulator
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Capacitance Definition
Q CV

• Take two chunks of conductor
• Separated by insulator
• Apply a potential V between them
• Charge will appear on the conductors, with Q
  CV on the higher-potential and Q- -CV on the
  lower potential conductor
• C depends upon both the geometry and the nature
  of the material that is the insulator

Q- -CV
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Calculating Capacitance?

• C f(geometry, dielectric)
• e.g. C eArea/separation eA/d for a
  parallel-plate capacitor
• With much symmetry, C can be calculated
• And capacitors are often manufactured in simple
  geometries!
• Without such symmetry approximation and
  estimation is necessary
• Can be made arbitrarily accurate
• Remember Laplace and field plotting?
• Tackle calculation, then estimation

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Example 1 Parallel-Plate Capacitor

1. Calculate field strength E as a function of
   charge Q on the plates
2. Integrate field to calculate potential V between
   the plates
3. QCV, C Q/V

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Example 1 Parallel-Plate Capacitor

• Gausss Law D, E ¹ 0 only on bottom face
• Charge enclosed AGQ/A

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Example 1 Parallel-Plate Capacitor
Q
Area A
e
Area A
-Q
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Example 2 Cylindrical Capacitor

• Two concentric cylindrical conductors, overlap
  length L
• e.g. co-axial TV lead cable
• Separated by a dielectric (insulator)

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Example 2 Cylindrical Capacitor
E
V Q
0V, -Q
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Estimating Capacitance

• When the electrodes are not as symmetrical as
  these examples
• Also our ideal parallel-plate capacitor
  should really look thus-

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Estimating Capacitance Principle

• Sketch equipotentials and field lines using field
  plotting
• Can be arbitrarily accurate
• More accuracy means moreV0 ¼ (V1V2V3V4)
• Use a computer!

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Estimating capacitance
eaxa a
ea
?7ea
? e7x3a
? e7x3a 2
CeA d
CTOTe7 F/unit depth 2
Equipotentials
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Estimating capacitance
ewxd h
ea
?7ea
? e7x3a
? e7x3a 2
CeA d
CTOTe7 F/unit depth 2
Equipotentials
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Estimating capacitance
Capacitance/unit depth (a)?
Capacitance?
CeA d
eax0.5a 0.5a
ea
e
Square
CeA d
eaxa a
ea
e
Square
CeA d
eax2a 2a
ea
e
Square
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Estimating capacitance
Capacitance/unit depth (a)?
Capacitance?
CeA d
eaxa a
ea
e
Square
CeA d
eaxa a
ea
e
Square
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Underlying Idea
C eA/d e5x2x/2x C 5ex Is 100 equivalent
to
Each C eA/d ex 10 in parallel,2 in
series CTOT 10ex/2 5ex
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Underlying Idea
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Estimation Example
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Estimation Example
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Estimation Example
Each of these is eA/d exx/x 4 in series, 30
in parallel Capacitance 30xe/4 Or
capacitance/unit depth 10e/4
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Estimating Capacitance Recipe

• Draw equipotentials as accurately as you have
  time for
• Using field mapping in reality
• Draw field lines to make square cells (cubes in
  3D)
• Field line and equipotentials cross at 90
• Make cells as square as possible
• Count series and parallel each is a capacitance
  of ex (e per unit depth when using a 2D diagram)