Capacitance

1
Lecture 6

• Capacitance
• Electric Current
• Circuits
• Resistance and Ohms law

2
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

3
More About 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

4
Fig. 16-19, p.551
5
Fig. 16-20, p.552
6
Capacitors in Series, cont

• The equivalent capacitance of a series
  combination is always less than any individual
  capacitor in the combination
• Demo

7
Fig. P16-34, p.564
8
Fig. P16-35, p.564
9
Problem-Solving Strategy

• Be careful with the choice of units
• Combine capacitors following the formulas
• 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

10
Problem-Solving Strategy, cont

• Combining 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

11
Problem-Solving Strategy, final

• Repeat the process until there is only one single
  equivalent capacitor
• A complicated circuit can often be reduced to one
  equivalent capacitor
• Replace capacitors in series or parallel with
  their equivalent
• Redraw the circuit and continue
• 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

12
Problem-Solving Strategy, Equation Summary

• Use the following equations when working through
  the circuit diagrams
• Capacitance equation C Q / DV
• Capacitors in parallel Ceq C1 C2
• Capacitors in parallel all have the same voltage
  differences as does the equivalent capacitance
• Capacitors in series 1/Ceq 1/C1 1/C2
• Capacitors in series all have the same charge, Q,
  as does their equivalent capacitance

13
Fig. 16-21, p.553
14
Fig. P16-57, p.566
15
Energy Stored in a Capacitor

• Energy stored ½ Q ?V
• From the definition of capacitance, this can be
  rewritten in different forms

16
Fig. 16-22, p.554
17
Applications

• Defibrillators
• When fibrillation occurs, the heart produces a
  rapid, irregular pattern of beats
• A fast discharge of electrical energy through the
  heart can return the organ to its normal beat
  pattern
• In general, capacitors act as energy reservoirs
  that can slowly charged and then discharged
  quickly to provide large amounts of energy in a
  short pulse

18
Capacitors with Dielectrics

• A dielectric is an insulating material that, when
  placed between the plates of a capacitor,
  increases the capacitance
• Dielectrics include rubber, plastic, or waxed
  paper
• C ?Co ?eo(A/d)
• The capacitance is multiplied by the factor ?
  when the dielectric completely fills the region
  between the plates

19
Capacitors with Dielectrics
20
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

21
An Atomic Description of Dielectrics

• Polarization occurs when there is a separation
  between the centers of gravity of its negative
  charge and its positive charge
• In a capacitor, the dielectric becomes polarized
  because it is in an electric field that exists
  between the plates

22
More Atomic Description

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

23
Fig. 16-30, p.560
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Table 16-1, p.557
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Fig. 16-1, p.532
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Fig. 16-23, p.557
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Fig. 16-26, p.558
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Fig. 16-28, p.560
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Fig. 16-29a, p.560
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Fig. 16-29b, p.560
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(No Transcript)
32
Electric Current

• Whenever electric charges of like signs move, an
  electric current is said to exist
• The current is the rate at which the charge flows
  through this surface
• Look at the charges flowing perpendicularly to a
  surface of area A
• The SI unit of current is Ampere (A)
• 1 A 1 C/s

33
Electric Current, cont

• The direction of the current is the direction
  positive charge would flow
• This is known as conventional current direction
• In a common conductor, such as copper, the
  current is due to the motion of the negatively
  charged electrons
• It is common to refer to a moving charge as a
  mobile charge carrier
• A charge carrier can be positive or negative

34
Current and Drift Speed

• Charged particles move through a conductor of
  cross-sectional area A
• n is the number of charge carriers per unit
  volume
• n A ?x is the total number of charge carriers

35
Current and Drift Speed, cont

• The total charge is the number of carriers times
  the charge per carrier, q
• ?Q (n A ?x) q
• The drift speed, vd, is the speed at which the
  carriers move
• vd ?x/ ?t
• Rewritten ?Q (n A vd ?t) q
• Finally, current, I ?Q/?t nqvdA

36
Current and Drift Speed, final

• If the conductor is isolated, the electrons
  undergo random motion
• When an electric field is set up in the
  conductor, it creates an electric force on the
  electrons and hence a current

37
Charge Carrier Motion in a Conductor

• The zig-zag black line represents the motion of
  charge carrier in a conductor
• The net drift speed is small
• The sharp changes in direction are due to
  collisions
• The net motion of electrons is opposite the
  direction of the electric field Demo

38
Electrons in a Circuit

• The drift speed is much smaller than the average
  speed between collisions
• When a circuit is completed, the electric field
  travels with a speed close to the speed of light
• Although the drift speed is on the order of 10-4
  m/s the effect of the electric field is felt on
  the order of 108 m/s
• c 3 x 108 m/s

39
Meters in a Circuit Ammeter

• An ammeter is used to measure current
• In line with the bulb, all the charge passing
  through the bulb also must pass through the meter

40
p.578
41
Fig. A17-1, p.591
42
Meters in a Circuit Voltmeter

• A voltmeter is used to measure voltage (potential
  difference)
• Connects to the two ends of the bulb

43
Resistance

• In a conductor, the voltage applied across the
  ends of the conductor is proportional to the
  current through the conductor
• The constant of proportionality is the resistance
  of the conductor

44
Fig. 17-CO, p.568
45
Resistance, cont

• Units of resistance are ohms (O)
• 1 O 1 V / A
• Resistance in a circuit arises due to collisions
  between the electrons carrying the current with
  the fixed atoms inside the conductor

46
Georg Simon Ohm

• 1787 1854
• Formulated the concept of resistance
• Discovered the proportionality between current
  and voltages

47
Ohms Law

• Experiments show that for many materials,
  including most metals, the resistance remains
  constant over a wide range of applied voltages or
  currents
• This statement has become known as Ohms Law
• ?V I R
• Ohms Law is an empirical relationship that is
  valid only for certain materials
• Materials that obey Ohms Law are said to be ohmic