Electric Energy

1
Chapter 16

• Electric Energy
• and
• Capacitance

2
Electric Potential Energy

• The electrostatic force is a conservative force
• It is possible to define an electrical potential
  energy function with this force
• Work done by a conservative force is equal to the
  negative of the change in potential energy

3
Work and Potential Energy

• There is a uniform field between the two plates
• As the charge moves from A to B, work is done on
  it
• W Fdq Ex (xf xi)
• ?PE - W
• - q Ex (xf xi)
• only for a uniform field

4
Potential Difference

• The potential difference between points A and B
  is defined as the change in the potential energy
  (final value minus initial value) of a charge q
  moved from A to B divided by the size of the
  charge
• ?V VB VA ?PE / q
• Potential difference is not the same as potential
  energy

5
Potential Difference, cont.

• Another way to relate the energy and the
  potential difference ?PE q ?V
• Both electric potential energy and potential
  difference are scalar quantities
• Units of potential difference
• V J/C
• A special case occurs when there is a uniform
  electric field
• DV VB VA -Ex Dx
• Gives more information about units N/C V/m

6
Energy and Charge Movements

• A positive charge gains electrical potential
  energy when it is moved in a direction opposite
  the electric field
• If a charge is released in the electric field, it
  experiences a force and accelerates, gaining
  kinetic energy
• As it gains kinetic energy, it loses an equal
  amount of electrical potential energy
• A negative charge loses electrical potential
  energy when it moves in the direction opposite
  the electric field

7
Energy and Charge Movements, cont

• When the electric field is directed downward,
  point B is at a lower potential than point A
• A positive test charge that moves from A to B
  loses electric potential energy
• It will gain the same amount of kinetic energy as
  it loses in potential energy

8
Summary of Positive Charge Movements and Energy

• When a positive charge is placed in an electric
  field
• It moves in the direction of the field
• It moves from a point of higher potential to a
  point of lower potential
• Its electrical potential energy decreases
• Its kinetic energy increases

9
Summary of Negative Charge Movements and Energy

• When a negative charge is placed in an electric
  field
• It moves opposite to the direction of the field
• It moves from a point of lower potential to a
  point of higher potential
• Its electrical potential energy increases
• Its kinetic energy increases
• Work has to be done on the charge for it to move
  from point A to point B

10
Electric Potential of a Point Charge

• The point of zero electric potential is taken to
  be at an infinite distance from the charge
• The potential created by a point charge q at any
  distance r from the charge is
• A potential exists at some point in space whether
  or not there is a test charge at that point

11
Electric Field and Electric Potential Depend on
Distance

• The electric field is proportional to 1/r2
• The electric potential is proportional to 1/r

12
Electric Potential of Multiple Point Charges

• Superposition principle applies
• The total electric potential at some point P due
  to several point charges is the algebraic sum of
  the electric potentials due to the individual
  charges
• The algebraic sum is used because potentials are
  scalar quantities

13
Electrical Potential Energy of Two Charges

• V1 is the electric potential due to q1 at some
  point P
• The work required to bring q2 from infinity to P
  without acceleration is q2V1
• This work is equal to the potential energy of the
  two particle system

14
Notes About Electric Potential Energy of Two
Charges

• If the charges have the same sign, PE is positive
• Positive work must be done to force the two
  charges near one another
• The like charges would repel
• If the charges have opposite signs, PE is
  negative
• The force would be attractive
• Work must be done to hold back the unlike charges
  from accelerating as they are brought close
  together

15
Problem Solving with Electric Potential (Point
Charges)

• Draw a diagram of all charges
• Note the point of interest
• Calculate the distance from each charge to the
  point of interest
• Use the basic equation V keq/r
• Include the sign
• The potential is positive if the charge is
  positive and negative if the charge is negative

16
Problem Solving with Electric Potential, cont

• Use the superposition principle when you have
  multiple charges
• Take the algebraic sum
• Remember that potential is a scalar quantity
• So no components to worry about

17
Potentials and Charged Conductors

• Since W -q(VB VA), no work is required to
  move a charge between two points that are at the
  same electric potential
• W 0 when VA VB
• All points on the surface of a charged conductor
  in electrostatic equilibrium are at the same
  potential
• Therefore, the electric potential is a constant
  everywhere on the surface of a charged conductor
  in equilibrium

18
Example 1
A uniform electric field of magnitude 250 V/m is
directed in the positive x-direction. A 12-µC
charge moves from the origin to the point (x, y)
(20 cm, 50 cm). (a) What was the change in the
potential energy of this charge? (b) Through what
potential difference did the charge move?
19
Example 2
The potential difference between the accelerating
plates of a TV set is about 25 kV. If the
distance between the plates is 1.5 cm, find the
magnitude of the uniform electric field in the
region between the plates.
20
Conductors in Equilibrium

• The conductor has an excess of positive charge
• All of the charge resides at the surface
• E 0 inside the conductor
• The electric field just outside the conductor is
  perpendicular to the surface
• The potential is a constant everywhere on the
  surface of the conductor
• The potential everywhere inside the conductor is
  constant and equal to its value at the surface

21
The Electron Volt

• The electron volt (eV) is defined as the energy
  that an electron gains when accelerated through a
  potential difference of 1 V
• Electrons in normal atoms have energies of 10s
  of eV
• Excited electrons have energies of 1000s of eV
• High energy gamma rays have energies of millions
  of eV
• 1 eV 1.6 x 10-19 J

22
Equipotential Surfaces

• An equipotential surface is a surface on which
  all points are at the same potential
• No work is required to move a charge at a
  constant speed on an equipotential surface
• The electric field at every point on an
  equipotential surface is perpendicular to the
  surface

23
Equipotentials and Electric Fields Lines
Positive Charge

• The equipotentials for a point charge are a
  family of spheres centered on the point charge
• The field lines are perpendicular to the electric
  potential at all points

24
Equipotentials and Electric Fields Lines Dipole

• Equipotential lines are shown in blue
• Electric field lines are shown in red
• The field lines are perpendicular to the
  equipotential lines at all points

25
Application Electrostatic Precipitator

• It is used to remove particulate matter from
  combustion gases
• Reduces air pollution
• Can eliminate approximately 90 by mass of the
  ash and dust from smoke

26
Application Electrostatic Air Cleaner

• Used in homes to relieve the discomfort of
  allergy sufferers
• It uses many of the same principles as the
  electrostatic precipitator

27
Application Xerographic Copiers

• The process of xerography is used for making
  photocopies
• Uses photoconductive materials
• A photoconductive material is a poor conductor of
  electricity in the dark but becomes a good
  electric conductor when exposed to light

28
The Xerographic Process
29
Application Laser Printer

• The steps for producing a document on a laser
  printer is similar to the steps in the
  xerographic process
• Steps a, c, and d are the same
• The major difference is the way the image forms
  on the selenium-coated drum
• A rotating mirror inside the printer causes the
  beam of the laser to sweep across the
  selenium-coated drum
• The electrical signals form the desired letter in
  positive charges on the selenium-coated drum
• Toner is applied and the process continues as in
  the xerographic process

30
Example 3
Two point charges Q1 5.00 nC and Q2 -3.00 nC
are separated by 35.0 cm. (a) What is the
electric potential at a point midway between the
charges? (b) What is the potential energy of the
pair of charges? What is the significance of the
algebraic sign of your answer?
31
Example 4
An electron starts from rest 3.00 cm from the
center of a uniformly charged sphere of radius
2.00 cm. If the sphere carries a total charge of
1.00 10-9 C, how fast will the electron be
moving when it reaches the surface of the sphere?
32
Example 5
A small spherical object carries a charge of 8.00
nC. At what distance from the center of the
object is the potential equal to 100 V? 50.0 V?
25.0 V? Is the spacing of the equipotentials
proportional to the change in voltage?
33
Capacitance

• A capacitor is a device used in a variety of
  electric circuits
• The capacitance, C, of a capacitor is defined as
  the ratio of the magnitude of the charge on
  either conductor (plate) to the magnitude of the
  potential difference between the conductors
  (plates)

34
Capacitance, cont

• Units Farad (F)
• 1 F 1 C / V
• A Farad is very large
• Often will see µF or pF

35
Parallel-Plate Capacitor

• The capacitance of a device depends on the
  geometric arrangement of the conductors
• For a parallel-plate capacitor whose plates are
  separated by air

36
Parallel-Plate Capacitor, Example

• The capacitor consists of two parallel plates
• Each have area A
• They are separated by a distance d
• The plates carry equal and opposite charges
• When connected to the battery, charge is pulled
  off one plate and transferred to the other plate
• The transfer stops when DVcap DVbattery

37
Electric Field in a Parallel-Plate Capacitor

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

38
Applications of Capacitors Camera Flash

• The flash attachment on a camera uses a capacitor
• A battery is used to charge the capacitor
• The energy stored in the capacitor is released
  when the button is pushed to take a picture
• The charge is delivered very quickly,
  illuminating the subject when more light is needed

39
Applications of Capacitors Computers

• Computers use capacitors in many ways
• Some keyboards use capacitors at the bases of the
  keys
• When the key is pressed, the capacitor spacing
  decreases and the capacitance increases
• The key is recognized by the change in capacitance

40
Capacitors in 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 can be used to show the path of
  the real circuit

41
Capacitors in Parallel

• When capacitors are first connected in the
  circuit, electrons are transferred from the left
  plates through the battery to the right plate,
  leaving the left plate positively charged and the
  right plate 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

42
Capacitors in Parallel

• The total charge is equal to the sum of the
  charges on the capacitors
• Qtotal Q1 Q2
• The potential difference across the capacitors is
  the same
• And each is equal to the voltage of the battery

43
More About Capacitors in Parallel

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

44
Capacitors in Parallel, final

• Ceq C1 C2
• The equivalent capacitance of a parallel
  combination of capacitors is greater than any of
  the individual capacitors

45
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

46
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

47
Capacitors in Series, cont

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

48
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

49
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

50
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

51
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

52
Energy Stored in a Capacitor

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

53
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

54
Example 6
(a) How much charge is on each plate of a 6.00-µF
capacitor when it is connected to a 9.0-V
battery? (b) If this same capacitor is connected
to a 1.50-V battery, what charge is stored?
55
Example 7
The potential difference between a pair of
oppositely charged parallel plates is 400 V. (a)
If the spacing between the plates is doubled
without altering the charge on the plates, what
is the new potential difference between the
plates? (b) If the plate spacing is doubled while
the potential difference between the plates is
kept constant, what is the ratio of the final
charge on one of the plates to the original
charge?
56
Example 8
A 1-megabit computer memory chip contains many
60.0 10-15 F capacitors. Each capacitor has a
plate area of 21.0 10-12 m2. Determine the
plate separation of such a capacitor. (Assume a
parallel-plate configuration). The diameter of an
atom is on the order of 10-10 m 1 Å. Express
the plate separation in angstroms.
57
Example 9
A series circuit consists of a 0.050-µF
capacitor, a 0.100-µF capacitor, and a 400-V
battery. Find the charge (a) on each of the
capacitors and (b) on each of the capacitors if
they are reconnected in parallel across the
battery.
58
Example 10
Three capacitors, C1 5.00 µF, C2 4.00 µF, and
C3 9.00 µF, are connected together. Find the
effective capacitance of the group (a) if they
are all in parallel, and (b) if they are all in
series.
59
Example 11
Four capacitors are connected as shown in the
figure below. (a) Find the equivalent capacitance
between points a and b. (b) Calculate the charge
on each capacitor if a 15.0-V battery is
connected across points a and b.
60
Example 12
Two capacitors, C1 25.0 µF and C2 5.00 µF,
are connected in parallel and charged with a
100-V power supply. (a) Calculate the total
energy stored in the two capacitors. (b) What
potential difference would be required across the
same two capacitors connected in series in order
that the combination store the same energy as in
(a)?
61
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

62
Capacitors with Dielectrics
63
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

64
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

65
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

66
Example 13
A capacitor with air between its plates is
charged to 100 V and then disconnected from the
battery. When a piece of glass is placed between
the plates, the voltage across the capacitor
drops to 25 V. What is the dielectric constant of
the glass? (Assume the glass completely fills the
space between the plates.)
67
Example 14
Two parallel plates, each of area 2.00 cm2, are
separated by 2.00 mm with purified nonconducting
water between them. A voltage of 6.00 V is
applied between the plates. Calculate (a) the
magnitude of the electric field between the
plates, (b) the charge stored on each plate, and
(c) the charge stored on each plate if the water
is removed and replaced with air.