Lecture 4 Capacitance and Capacitors

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Lecture 4Capacitance and Capacitors
Chapter 16.6 ? 16.10
Outline

• Definition of Capacitance
• Simple Capacitors
• Combinations of Capacitors
• Capacitors with Dielectrics

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Capacitance
History
Introduction
General definition
The capacitance C of a capacitor is the ratio of
the charge (Q) on either conductor plate to the
potential difference (?V) between the plates.
Units of capacitance are farads (F) 1F ? 1C/1V
Q C ? ? ?V
C(Earth) 1 F C(adult) 150 pF 150 10?12 F
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The Parallel-Plate Capacitor
The capacitance of a parallel-plate capacitor
whose plates are separated by air is
A is the area of one of the plates d is the
distance between the plates ?0 is permittivity of
free space
A C ?0 ? d
More about capacitors
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Capacitors
Problem A parallel-plane capacitor has an area
of A5cm2 and a plate separation of d5mm. Find
its capacitance.
Unit conversion A 5 cm2 5 10?4 m2
d 5 mm 5 10?3
m C ?0 A/d 8.85 10?12 C2 / (N m2) 5 10?4 m2 /
5 10?3 m 8.85 10?13 C2/(N m) 8.85 10?13 F
0.885 pF N/C V/m ? C/N m/V, FC/V C2/(N m)C
(C/N)/m C (m/V)/m C/V F
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Combinations of Capacitors
In real electric circuits capacitors can be
connected in various ways. In order to design a
circuit with desired capacitance, equivalent
capacitance of certain combinations of capacitors
can be calculated.

• There are 2 typical combinations of capacitors
• Parallel combination
• Series combination

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Parallel Combination
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Parallel Combination

• The left plate of each capacitor is connected to
  the positive terminal of a battery by a wire ?
• the left plates are at the same potential ?
• the potential differences across the capacitors
  are the same, equal to the voltage of the battery
  (?V).
• The charge flow ceases when the voltage across
  the capacitors equals to that of the battery and
  the capacitors reach their maximum charge.

Examples
Q Q1 Q2 Q1 C1 ?V Q2 C2 ?V
Q Ceq ?V Ceq ?V C1 ?V C2 ?V Ceq C1 C2
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Series Combination
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Series Combination
The magnitude of the charge is the same on all
the plates. The equivalent capacitor must have a
charge Q on the right plate and Q on the left
plate.
Q ?V ? Ceq
Examples

• Q Q Q
• ? ?
• Ceq C1 C2

• 1 1 1
• ? ?
• Ceq C1 C2

?V ?V1 ?V2 ?V1 Q/C1 ?V2 Q/C2
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Energy Stored in a Capacitor
The work required to move a charge ?Q through a
potential difference ?V is ?W ?V ?Q.
?V Q/C, Q is the total charge on the
capacitor. The voltage on the capacitor linearly
increases with the magnitude of the
charge. Additional work increases the energy
stored.
W ½ Q ?V ½ (C ?V) ?V ½C (?V)2 Q2/2C
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Capacitors with Dielectrics
A dielectric is an insulating material. The
dielectric filling the space between the plates
completely increases the capacitance by the
factor ? gt 1, called the dielectric constant.
If ?V0 is the potential difference (voltage)
across a capacitor of a capacitance C0 and a
charge Q0 in the absence of a dielectric.
Filling the capacitor with a dielectric reduces
the voltage by the factor ? to ?V, so that ?V
?V0/?.
C Q0/?V Q0/?V0/? ? Q0/?V0 ? C0
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Dielectric Strength
For a parallel-plate capacitor C ? ?0 A/d
The formula shows that the capacitance can be
made very large by decreasing the plate
separation. In practice, the lowest value of d is
limited by the electric discharge through the
dielectric. The discharge occurs when the
electric field in the dielectric material reaches
its maximum, called dielectric strength. Dielectri
c strength of air is 3 106 V/m.
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Summary

• Capacitance is defined as the charge over the
  potential difference
• Capacitance of parallel-plate capacitor is
  directly proportional to the plate area and
  inversely proportional to the plate separation
• The equivalent capacitance of a parallel
  combination of capacitors equals to the sum of
  individual capacitances
• The inverse equivalent capacitance of a series
  combination of capacitors equals to the sum of
  the inverse individual capacitances
• Placing a dielectric between the plates of a
  capacitor increases the capacitance by a factor
  ?, called the dielectric constant