TOPIC 5 Capacitors and Dielectrics

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TOPIC 5Capacitors and Dielectrics
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Capacitors

• Capacitors are a means of storing electric charge
  (and electric energy)
• It takes energy to bring charge together
• A capacitor allows more charge to be stored for a
  given energy
• It does this by reducing the potential at which
  the charge is stored
• It can do this by bringing an opposite charge
  into close proximity, to reduce the overall
  repulsion

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Capacitance
Capacitance (C) is charge per unit potential
difference C Q/V Unit is Farad (F) 1 F 1
Coulomb/Volt Typical capacitances measured in ?F
(106 F) or pF (1012 F)
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Parallel Plate Capacitor
Two plates, area A, separation d, carrying charge
?Q. Gausss Law (using dotted Gaussian surface
shown) E A Q/?0 ? E Q/?0 A
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Example 1 Parallel Plate Capacitor A parallel
plate capacitor has plates with dimensions 3 cm
by 4 cm, separated by 2 mm. The plates are
connected across a 60 V battery. Find (a) the
capacitance (b) the magnitude of charge on each
plate (c) the energy stored in the capacitor
see later!
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Example 2 Cylindrical Capacitor What is the
capacitance of a long cylindrical (coaxial) cable
of inner radius a, outer radius b and length L as
shown?
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Example 3 Spherical Capacitor What is the
capacitance of two concentric spherical
conducting shells of inner radius a and outer
radius b?
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Capacitors in Parallel
Capacitors connected as shown, with terminals
connected together, are said to be in
parallel. They behave as a single capacitor with
effective capacitance C. Total charge Q Q1 Q2
C1V C2V Therefore C Q/V C1 C2
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Capacitors in Series
Capacitors connected together as shown, sharing
one common terminal, are said to be in
series. They behave as a single capacitor with
effective capacitance C. The external charge
stored is Q. The voltages across the capacitors
Vi Q/Ci must add up to V. Therefore V Q/C1
Q/C2 Q/C
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Example 4 Capacitor Network If each of the
individual capacitors in the network below has a
capacitance C, what is the overall effective
capacitance?
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Energy stored in a Capacitor
Adding an increment of charge dq to a capacitor
requires work dW V dq q/C dq This is
obviously the increase in (potential) energy
stored of the capacitor U The total energy
required to charge a capacitor from zero charge
to Q is therefore Since Q C V, we can express
this in other ways
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Example 1 Parallel Plate Capacitor A parallel
plate capacitor has plates with dimensions 3 cm
by 4 cm, separated by 2 mm. The plates are
connected across a 60 V battery. Find (a) the
capacitance (b) the magnitude of charge on each
plate (c) the energy stored in the capacitor
see later! Previously C 5.3 pF Q 3.2?1010 C
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Energy stored in a Capacitor (2)
The energy stored in the capacitor can also be
considered as the energy stored in its electric
field. We have For the parallel plate capacitor
we also have V E d and So But A d is the
volume where the electric field exists, so the
energy density is This is a general result for
the energy density in a field.
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Dielectrics
A conductor contains free charges that can move
through the material. A dielectric contains bound
charges, which cannot move freely but will
displace through small distances when affected by
an electric field.
This leaves excess bound charges on the surface
of the material. This reduces the electric field
within the bulk of the material.
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Dielectrics and Capacitors
The factor by which the electric field is reduced
is known as the dielectric constant k (or ?r). If
the gap between the plates of a capacitor is
filled with dielectric material, the voltage
between the plates for a given charge will also
be reduced by the factor k. Since C Q / V, this
means that C is increased by k. For the parallel
plate capacitor, we therefore have
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Example 5 Demonstrate that the energy stored in a
spherical capacitor is consistent with an energy
density stored in the field of