Capacitance

1
Chapter 26

• Capacitance
• and
• Dielectrics

2
Capacitance

• The capacitance, C, is a measure of the amount of
  electric charge stored (or separated) for a given
  electric potential
• SI unit Farad (F) 1 F 1 C / V
• A 1 Farad capacitance is very large µF or pF
  capacitances are more common

3
Capacitor

• A capacitor is a device used in a variety of
  electric circuits to store electric charge
• Capacitance of a capacitor is the ratio of the
  magnitude of the charge on either conductor
  (plate) to the magnitude of the potential
  difference between the conductors (plates)
• This capacitance of a device depends on the
  geometric arrangement of the conductors

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Parallel-Plate Capacitor

• This capacitor consists of two parallel plates
  (each of area A) separated by a distance d each
  carrying equal and opposite charges
• Each plate is connected to a terminal of the
  battery
• The battery is a source of potential difference
• If the capacitor is initially uncharged, the
  battery establishes an electric field in the
  connecting wires

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Parallel-Plate Capacitor

• The field applies a force on electrons in the
  wire just outside of the plates, causing the
  electrons to move onto the negative plate until
  equilibrium is achieved
• The plate, the wire and the terminal are all at
  the same potential, and there is no field present
  in the wire and the movement of the electrons
  ceases
• At the other plate, electrons are moving away
  from the plate and leaving it positively charged

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Parallel-Plate Capacitor

• The electric field due to one plate
• The total electric field between the plates is
  given by
• The field outside the plates is zero

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Parallel-Plate Capacitor

• In the final configuration, the potential
  difference across the capacitor plates is the
  same as that between the terminals of the battery
• For a parallel-plate capacitor whose plates are
  separated by air

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Electric Field in a Parallel-Plate Capacitor

• The electric field between the plates is uniform
  near the center and nonuniform near the edges
• The field may be taken as constant throughout the
  region between the plates

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Spherical Capacitor
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Chapter 26Problem 5

• An air-filled capacitor consists of two parallel
  plates, each with an area of 7.60 cm2, separated
  by a distance of 1.80 mm. A 20.0-V potential
  difference is applied to these plates. Calculate
  (a) the electric field between the plates, (b)
  the surface charge density, (c) the capacitance,
  and (d) the charge on each plate.

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Electric Circuits

• A circuit is a collection of objects usually
  containing a source of electrical energy (such as
  a battery) connected to elements that convert
  electrical energy to other forms
• A circuit diagram a simplified representation
  of an actual circuit is used to show the path
  of the real circuit
• Circuit symbols are used to represent various
  elements (e.g., lines are used to represent
  wires, batterys positive terminal is indicated
  by a longer line)

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Capacitors in Parallel

• When capacitors are first connected in parallel
  in the circuit, electrons are transferred from
  the left plates through the battery to the right
  plates, leaving the left plates positively
  charged and the right plates negatively charged
• The flow of charges ceases when the voltage
  across the capacitors equals that of the battery
• The capacitors reach their maximum charge when
  the flow of charge ceases

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Capacitors in Parallel

• The total charge is equal to the sum of the
  charges on the capacitors Qtotal Q1 Q2
• The potential differences across the capacitors
  is the same and each is equal to the voltage of
  the battery
• A circuit diagram for two
• capacitors in parallel

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Capacitors in Parallel

• The capacitors can be replaced with one capacitor
  with a equivalent capacitance Ceq the
  equivalent capacitor must have exactly the same
  external effect on the circuit as the original
  capacitors

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Capacitors in Parallel

• For more than two capacitors in parallel
• The equivalent capacitance of a parallel
  combination of capacitors is greater than any of
  the individual capacitors

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Capacitors in Series

• When a battery is connected to the circuit,
  electrons are transferred from the left plate of
  C1 to the right plate of C2 through the battery
• As this negative charge accumulates on the right
  plate of C2, an equivalent amount of negative
  charge is removed from the left plate of C2,
  leaving it with an excess positive charge
• All of the right plates gain charges of Q and
  all the left plates have charges of Q

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Capacitors in Series

• An equivalent capacitor can be found that
  performs the same function as the series
  combination
• The potential differences add up to the battery
  voltage

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Capacitors in Series

• For more than two capacitors in series
• The equivalent capacitance is always less than
  any individual capacitor in the combination

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Problem-Solving Strategy

• Be careful with the choice of units
• Combine capacitors
• When two or more unequal capacitors are connected
  in series, they carry the same charge, but the
  potential differences across them are not the
  same
• The capacitances add as reciprocals and the
  equivalent capacitance is always less than the
  smallest individual capacitor

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Problem-Solving Strategy

• Be careful with the choice of units
• Combine capacitors
• When two or more capacitors are connected in
  parallel, the potential differences across them
  are the same
• The charge on each capacitor is proportional to
  its capacitance
• The capacitors add directly to give the
  equivalent capacitance

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Problem-Solving Strategy

• Redraw the circuit and continue
• Repeat the process until there is only one single
  equivalent capacitor
• To find the charge on, or the potential
  difference across, one of the capacitors, start
  with your final equivalent capacitor and work
  back through the circuit reductions

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Equivalent Capacitance
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Chapter 26Problem 25

• Find the equivalent capacitance between points a
  and b in the combination of capacitors shown in
  the figure

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Energy Stored in a Capacitor

• Before the switch is closed, the energy is stored
  as chemical energy in the battery
• When the switch is closed, the energy is
  transformed from chemical to electric potential
  energy
• The electric potential energy is related to the
  separation of the positive and negative charges
  on the plates
• A capacitor can be described as a device that
  stores energy as well as charge

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Energy Stored in a Capacitor

• Assume the capacitor is being charged and, at
  some point, has a charge q on it
• The work needed to transfer a charge from one
  plate to the other
• The total work required
• The work done in charging the capacitor appears
  as electric potential energy U

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Energy Stored in a Capacitor

• This applies to a capacitor of any geometry
• The energy stored increases as the charge
  increases and as the potential difference
  increases
• The energy can be considered to be stored in the
  electric field
• For a parallel-plate capacitor, the energy can be
  expressed in terms of the field as U ½ (eoAd)E2

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Capacitors with Dielectrics

• A dielectric is an insulating material (e.g.,
  rubber, plastic, etc.)
• When placed between the plates of a capacitor, it
  increases the capacitance C ? Co ? eo (A/d)
• ? - dielectric constant
• The capacitance is
• multiplied by the factor ?
• when the dielectric
• completely fills the region
• between the plates

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Capacitors with Dielectrics

• Tubular metallic foil interlaced with thin
  sheets of paraffin-impregnated paper rolled into
  a cylinder
• Oil filled (for high-V capacitors) interwoven
  metallic plates are immersed in silicon oil
• Electrolytic (to store large amounts of charge at
  relatively low voltages) electrolyte is a
  solution that conducts electricity via motion of
  ions in the solution

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Dielectric Strength

• For any given plate separation, there is a
  maximum electric field that can be produced in
  the dielectric before it breaks down and begins
  to conduct
• This maximum electric field is called the
  dielectric strength

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Chapter 26Problem 37

• Determine (a) the capacitance and (b) the maximum
  voltage that can be applied to a Teflon-filled
  parallel-plate capacitor having a plate area of
  175 cm2 and an insulation thickness of 0.040 0 mm.

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Electric Dipole

• An electric dipole consists of two charges of
  equal magnitude and opposite signs separated by
  2a
• The electric dipole moment p is directed along
  the line joining the charges from q to q and
  has a magnitude of p 2aq
• Assume the dipole is placed in a uniform field,
  external to the dipole (it is not the field
  produced by the dipole) and makes an angle ? with
  the field
• Each charge has a force of F Eq acting on it

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Electric Dipole

• The net force on the dipole is zero
• The forces produce a net torque on the dipole
• t 2Eqa sin ? pE sin ?
• The torque can also be expressed as the cross
  product of the moment and the field
• The potential energy can be expressed as

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An Atomic Description of Dielectrics

• Molecules are said to be polarized when a
  separation exists between the average position of
  the negative charges and the average position of
  the positive charges
• Polar molecules are those in which this condition
  is always present
• Molecules without a permanent polarization are
  called nonpolar molecules
• The average positions of the positive and
  negative charges act as point charges, thus polar
  molecules can be modeled as electric dipoles

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An Atomic Description of Dielectrics

• A linear symmetric molecule has no permanent
  polarization (a)
• Polarization can be induced by placing the
  molecule in an electric field (b)
• Induced polarization is the effect that
  predominates in most materials used as
  dielectrics in capacitors

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An Atomic Description of Dielectrics

• The molecules that make up the dielectric are
  modeled as dipoles in the absence of an electric
  field they are randomly oriented
• An external electric field produces a torque on
  the molecules partially aligning them with the
  electric field

37
An Atomic Description of Dielectrics

• The presence of the positive (negative) charge on
  the dielectric effectively induces some of the
  negative (positive) charge on the metal
• This allows more charge on the plates for a given
  applied voltage and the capacitance increases

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• Answers to Even Numbered Problems
• Chapter 26
• Problem 12
• 17.0 µF
• 9.00V
• 45.0 µC and 108 µC

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• Answers to Even Numbered Problems
• Chapter 26
• Problem 36
• 13.3 nC
• 272 nC

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Answers to Even Numbered Problems Chapter 26
Problem 50 (b) 40.0 µF (c) 6.00 V across 50
µF with charge 300 µF 4.00 V across 30 µF with
charge 120 µF 2.00 V across 20 µF with charge 40
µF 2.00 V across 40 µF with charge 80 µF