Electrical Energy and Capacitance

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Electrical Energy and Capacitance

• Chapter 25 Tipler

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Assembly of a system of charges

• We know that Vkq/r.
• To bring a charge in from infinity requires work
  to be done
• Work ?U qV

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Assembly of a system of charges

• W2q2V1kq2q1/R1,2
• W3q3V1 q3V2 kq3q1/R1,3 kq3q2/R2,3
• And so UsysW2W3

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Energy of a system of charges

• With a bit of mathematical trickery we get

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Consider a spherical conductor
Same result as for a collection of point
charges! Good for any shape!
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Capacitance spherical conductor
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How big is a Farad?

• Find the radius of a spherical conductor with a
  capacitance of 1 Farad.
• Thats 1400 times the radius of Earth!

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Capacitance and Charge?

• Q A sphere of capacitance C1 has a charge of
  20µC. If the charge is increased to 60 µC, what
  is the new capacitance?
• A C1 Capacitance is independent of charge
  because if the charge is tripled, potential is
  also tripled. C depends only on the radius of the
  sphere!

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Parallel Plate Capacitors

• Each plate contributes
• So the total field is

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Remember your eLab?

• We saw a linear relationship between C and A.
• We saw an inverse relationship between C and d.

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Try this!

• A parallel plate capacitor has square plates 10
  cm on each side that are separated by 1 mm.
• calculate C
• If this capacitor is charged to 12 V, how much
  charge is transferred from one plate to the other?

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Solution
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Cylindrical Capacitors

• Q is placed on the outer cylinder while -Q
  is placed on the inner cylinder.
• Since E depends on r, you must integrate to find
  C.

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Whats going on in there?

• Charge is transferred from one plate to another.
• Work is done.
• Each charge experiences a change in its energy
  (U).

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Moving charge

• When a small positive charge, dq, is moved from
  the plate to the plate, its potential energy
  is increased by dUV dq.

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Total increase in Potential Energy
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Potential Energy stored in a Capacitor

• Using C Q/V, we can express U in many ways

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Energy density

• Using the equation for electrostatic potential
  energy for a capacitor

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Combinations of Capacitors

• Parallel Capacitors
• Upper plates of both are at a common potential
  lower plates at common potential.
• Same potential difference across both capacitors.

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Combinations of Capacitors

• Series Capacitors
• The magnitude of the charge on both capacitors
  must be the same.
• ?V across both capacitors is equal to the sum of
  the ?V across each capacitor.

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Equivalent Capacitance

• Parallel capacitors
• Ceq C1 C2 C3
• Qtotal Q1 Q2 Q3

• Series Capacitors

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Practice makes Perfect!

• Homework Chapter 25 12 23, 27, 29.