Physics 2120 University Physics

1
Physics 2120University Physics

• Dr. Bill Robertson
• Middle Tennessee State University

2
Course Overview

• Text Fundamentals of Physics, 8th Edition by
  Walker, Halliday, and Resnick.
• Web site www.mtsu.edu/wroberts
• Office WPS 207
• Office Hours Monday Wednesday mornings
• Grading
• 3 Exams (In-class 3 worth 20, 20, 20) 60
• Homework 10
• Final Exam 30

3
Electrostatics and charge

• How was the concept of electric charge
  discovered? History Earliest experiments on
  charge involved simple static attraction and
  repulsion effects with dielectric rods rubbed on
  cloth or fur. From these experiments a number of
  basic terms and a law of attraction (Coulombs
  law) was derived.

4
Charge

• The electrostatic effect were attributed to the
  fact that objects could become charged.
• There were two types of charge positive and
  negative.
• Like charges repel and opposite charges attract.
• Charge is measured in Coulombs
• Smallest unit of charge, e 1.6 x 10-19 C

5
Conductors and insulators

• In general terms materials fall into the category
  of
• Insulators materials in which charge cannot move
  freely (rubber, plastic, pure water..)
• Conductors materials in which charge can move
  freely (metals, water with dissolved impurities,
  bunnies)
• We now understand that conduction occurs because
  of the movement of electrons (negative charges).

6
Example

• Induced charge A charged insulating rod held
  close to one end of a neutral metal rod attracts
  the metal rod. Why?

7
Save the bunny

• The physics is all wrong, but
• http//www.youtube.com/watch?v3K1YznzxQmw

8
Coulombs law

• Charges were observed to attract or repel,
  Coulombs law quantified the size and direction
  of the force between charges.
• Similar to Newtons gravitational law, inverse
  square dependence on distance.
• k 8.99x109 Nm2/C2 1/4peo

9
Example

• What is the force on an electron situated 1 cm
  away from a 0.1 C positive charge?

10
Example (leave the math!)

• Two otherwise identical metal spheres carry
  charges Q1 and Q2.  When their centers are a
  distance 1.19 m apart, the spheres attract one
  another with a force of magnitude 0.0853 N.  The
  spheres are then briefly put into contact with
  one another, and again separated to a distance of
  1.19 m.  The spheres are then found to repel one
  another with a force of magnitude 0.0196 N.  What
  were the original charges on the spheres if
  sphere 1 was initially positive?

11
Example

• Three charges Q1, Q2, and Q3 are attached to the
  corners of an equilateral triangle of side s
  0.50 m.  Find the net electrical force acting on
  the charge Q2 if Q1 2.0 mC, Q2 7.0 mC, and Q3
  4.0 mC.

12
Circuits

• The practical side of the electromagnetism.
• We will cover some basic concepts that are
  required for the lab work. We will define these
  terms and then reconcile more completely with the
  Coulomb force and electric field material as the
  semester progresses.

13
Current

• Current is the rate of flow of charge past a
  point in space (e.g. in a wire or a beam of
  charged particles).
• Typically given the symbol i or I.
• Mathematically
• Unit of current is the Ampere (amp) 1 Amp 1 C/s
• Current can flow in a conductor but not an
  insulator
• If the current is constant and 5 C passes through
  a wire in 10 seconds what is I?

14
EMF and Voltage

• In order to get charges to move around a circuit
  there needs to be some force that causes them
  to move. (EMF stands for Electromotive Force, an
  olde worlde term no longer popular).
• An EMF source maintains a potential energy
  difference between its terminals.
• The EMF source of initial interest is the
  battery. More generally called a dc-power supply.
• dc direct current as opposed to ac alternating
  current

15
EMF and Voltage

• EMF is defined in terms of the work done on a
  charge that travels from one terminal to the
  other.
• EMF is defined as the work done per unit charge.
• Units of EMF are work/charge i.e. Joules/Coulomb
  Volt (1 Volt 1 Joule per Coulomb)

16
Examples

• How much work is done when a steady current of 5
  mA flows for 3 minutes out of a 9 Volt battery?
• What is the instantaneous power expended by the
  battery during this time?
• Review Power Work done per unit time and is
  measured in Watts.

17
Energy measured in electron volts
Power from a battery

• Power from batteryVoltage x Current

• The unit of the Volt is energy per coulomb thus
  we can express energy as charge times voltage.
• When a an electron moves through a voltage of 1
  volt it acquires an increase in energy of 1
  electron volt.

18
Charge accelerator

• Electrons are accelerated (in vacuum) by a
  voltage of 20 kV. What is the energy of an
  electron in electron volts and in joules? How
  fast is the electron traveling?

19
Resistance

• We have said that there are two classes of
  materials conductors and insulators.
• A perfect conductor lets charge flow freely
  whereas an insulator will not let charge flow at
  all.
• In practice materials, and their geometry (e.g.
  thin wire versus thick wire), offer some
  impediment to the flow of charge. This property
  is known as resistance.

20
Resistance and Ohms Law

• A perfect conductor would have zero resistance
• A perfect insulator would have infinite
  resistance
• How is resistance defined?
• This relation is known as Ohms law. (I tend to
  remember it as VIR)
• The unit of resistance is the Ohm (W) (1 volt per
  amp equal 1 Ohm)

21
Circuit Symbols

• The resistor
• The battery (positive terminal long line,
  negative short line). Current always flows from
  positive to negative.
• The ground connection

22
Example

• A 9 V battery is attached to a 250 Ohm resistor.
  What current flows in the circuit?
• Circuit diagrams Which is the positive terminal
  of the battery? Which direction does current
  flow?

23
Example

• A 15V battery is connected end-to-end with a
  1.2 kW resistor.  The negative terminal of the
  battery is grounded.
• (a) Draw the circuit diagram, label the circuit
  elements, and draw in and label the current.
• (b) Wires used to connect circuit elements have a
  negligible resistance within a circuit (compared
  to other resistances in that circuit). Discuss
  the consequences of this fact when connecting a
  circuit and analyzing its behavior.
• (c) Find the value of the current in the circuit.
• (d) Plot a graph of Voltage vs. Position around
  the circuit
• (e) Plot a graph of Current vs. Position around
  the circuit

24
Solving single loop circuits

• Rule 1 The sum of the potential around a closed
  loop of any circuit must equal zero.
• Across a battery the potential change is positive
  from the negative to the positive terminal
  negative otherwise.
• The potential drop across a resistor is negative
  if the path goes in the same direction as the
  current across the resistor positive otherwise.
• Rule 2 The sum of the currents into a junction
  equals the sum of currents out of a junction.

25
Example

• Solve for the current in the circuit below.

26
Resistors in parallel and in series

• What is the one resistance that could replace the
  two resistors in the following circuits?
• What is common (current or voltage) for the (a)
  series and (b) parallel resistors?

27
Combining resistors summary

• Resistors in parallel
• Resistors in series

28
Solving circuits

• If possible see if the resistors can be combined
  using the parallel and series combination rules.
  Simplifying the circuit by replacing parallel and
  series resistors by their equivalent values can
  allow the circuit parameters to be determined.

29
Example

• Solve the total current drawn from the battery
  and the current in R3. R1 R3 30W, R2 40 W
  R4 60 W. V 12 V.

30
Real batteriesinternal resistance

• Ideally a battery has a given EMF value. When
  current is drawn from a real battery the voltage
  between the terminals falls from the ideal EMF
  value. The larger the current the more the
  reduction from the ideal EMF value. This effect
  is modeled as an internal series resistance
  within the battery.

31
Real batteries

• A 9 V battery gives a voltage of 8.7 V when a
  current of 250 mA is drawn from it. What is the
  internal resistance of the battery? What voltage
  would there be if 25 mA were drawn from the
  battery?

32
Multiple loop circuits

• Apply the loop potential rule around all the
  loops of the circuit. Each loop will give one
  equation relating the potentials and the loop
  currents. Use simultaneous equations to solve for
  the quantity of interest.

33
Example

• Find the value of the current and its direction
  for each resistor in the circuit shown below.
• (b) What would be the reading on a voltmeter if
  its black lead were connected to point A and its
  red lead were connected to point B?

V1 34 V  V2 20 V  R1 4.0 W         R2
7.0W           R3 10W   R4 9.0 W
34
Example for you to ponder

• What is the diagonal resistance of a cube of
  resistors each with resistance of 1 W?

35
Power dissipation in a resistor

• When a current I flows through a resistor what is
  the power generated?
• Every charge, q, changes its energy as it crosses
  the resistor by a value of q DV where DV is the
  voltage drop across the resistor. By Ohms law
  DVIR, and the charge that flows per second is I.
  Thus the energy change per second is I2R.
• Power dissipated I2R
• How else can we write this using Ohms Law?

36
Capacitance

• A capacitor is a device that stores charge.
• A capacitor consists of two conductors separated
  by an insulator.
• The simplest capacitor consists of a pair of
  parallel metal plates.
• When charge is moved from one plate to the other
  a voltage develops between the plates. Why?

37
Capacitance

• The voltage difference between the plates, V, and
  the charge on the plates,Q, are related by
• Q CV
• The units of capacitance are Coulombs per Volt
  which is named the Farad (1 F 1 C/V).
• Capacitance is determined by the geometry of the
  conductors (e.g. area of plates and their
  separation).

38
RC discharge

• Imagine a charged capacitor connected in series
  with a switch and a resistor. If the capacitor
  initially hold a charge Q0 ,what is capacitor
  charge Q as a function of time after the switch
  is closed?

39
Charging RC circuit

• We could similarly derive the equation for the
  charging of a capacitor as
• The quantity RC is called the time constant often
  given the symbol t. Note Ohms times Farads equals
  seconds

40
Energy stored in a capacitor

• A capacitor, C, that holds a charge Q stores
  potential energy U
• How else can this expression be written (remember
  QCV)

41
R1 100 kW, R2 50 kW, C 20 mF, Vin 50 V

• Consider the circuit shown in the diagram.  At
  t0 the switch is thrown to position A the
  capacitor is initially uncharged. (a) What is the
  current in the circuit immediately after the
  switch is closed? (b) At what time, t0, is the
  capacitor charge 90 of its maximum value? (c)
  What is the voltage across the capacitor at
  t0?(d) At what rate is energy being delivered by
  the voltage source at t0?

42
R1 100 kW, R2 50 kW, C 20 mF, Vin 50 V

• At time to, the switch is thrown to position B.
• (e) How long does it take after the switch is
  thrown to position B for the charge on the
  capacitor to be reduced by 90?  Call this time
  t1. (f) At what rate is energy being lost due to
  Joule heating at time t1?(g) At what rate is
  energy being delivered by the capacitor at time
  t1? (Does this result make sense?)

43
Electric Field

• The electric field is the force field that a
  single charge or a collection of charges creates
  in the surrounding space.
• The Electric field is defined as the force per
  unit charge i.e. the E-field at a point in space
  is equal to the force that a 1 Coulomb charge (a
  unit charge) would feel at that point.

44
E-field due to a point charge

• From Coulombs law the force between two charges
  is
• If one charge (say q2) is a unit charge then E
  due to q1 is defined as the force per unit charge

45
E-field is a vector field

• The electric field is a vector fieldit has a
  magnitude and a direction at every point.
• The E-field vector points away from a positive
  charge and towards a negative charge.
• The electric field at a point due to a collection
  of charges is the vector sum of the E-field
  created by every charge.
• A charge does not see its own E-field.

46
Field Line representation

• Maxwell came up with the field line
  representation as a way to visualize the electric
  field.
• The field lines show the direction of the E-field
  and the density of lines is related to the
  strength of the field.
• Field lines cannot cross!
• The field line model is a useful representation
  rather than a quantitative method

47
E-Field due to a Point Charge

• Drawing not so good!
• Lines should be uniformly spaced and emanate from
  the center of the charge

48
E-field due to a sheet of charge

• What will the E-field due to an infinite sheet of
  charge look like?
• We wont get a quantitative answer (yet) but we
  will figure out the direction of the field.

49
Example

• Find the field at a distance r along the x-axis
  of two opposite charges spaced by a distance d
  (see figure). Show that in the limit that rgtgtd
  the field is

50
Charge density distributions

• In many cases the charge is distributed along a
  line, over a surface, or in a volume.
• The charge is defined as a linear, areal, or
  volume density
• E.g. A 1 m2 surface has 0.01 C of charge evenly
  distributed. s0.01/1 C m-2
• The charge in an infinitesimal area dA sdA

51

• A line of charge of length L has a nonuniform
  linear charge density given by l Ax, where A is
  a constant.  The line is situated along the
  positive x-axis, as shown in the diagram below.
• (a) What are the SI units of the constant A?
• (b) Find an expression for the total line charge,
  Q.
• (c) Find an expression for the x-component of the
  electric field at the point P, a distance d from
  the end of the line.

52
Example

• What is the linear charge density?
• Calculate the E-field at a distance L
  perpendicular to the circle of charge, Q.
• In the limit of very large L does the result make
  sense?

53
Example

• Find the surface charge density and the E-field
  at the point A. Examine the case x?8. What is E
  as R? 8? (Infinite plane of charge).

54
Electric Field and Gausss Law

• Gausss law relates the flux of the electric
  field through a closed surface to the total
  charge enclosed.
• Define flux
• Define closed surface and surface integral
• Dielectric permittivity eo

55
Flux

• A measure of the flow of electric field through
  a surface.
• Analogy imagine the flow of water through wire
  loop. Can you see how the flow depends on the
  orientation of the loop and the direction of the
  flow of water? When is the flow the greatest?

56
Flux

• The electric field flux, fe, through a surface is
  given by the surface integral
• What is the flux through a 1 cm2 loop in a
  uniform E-field is the area of the loop is (a)
  parallel to E, (b) 45 degrees to E and (c) 90
  degrees to E?
• Note the direction of dA is outward from a closed
  surface

57
Example

• What is the flux through the shaded area in the
  figure due to an E-field given by

58
Gausss law for a point charge

• We know the E-field (direction and magnitude) for
  a point charge.
• Lets examine the E-field flux through a closed
  spherical surface that has the point charge Q at
  its center.
• What does Gausss law give for the value of E?

59
E-field due to a non-conducting sheet of charge

• Imagine a large non-conducting sheet of charge
  with surface charge density of s C m-2. Use
  Gausss law to find an expression for the E-field
  above and below the charged sheet.
• Consider a pill box Gaussian surface

60
E-field due to capacitor plates

• Capacitor plates (when charged) can be
  represented by two parallel sheets of opposite
  charge. Using the result of the E-field due to a
  single sheet of charge figure out the field above
  below and in between two capacitor plates.

61
E-field due to a charged conductor

• Inside a conductor the E-field must be zero. Why?
  Charges in a conductor are free to move, if there
  is an E-field they will move! They keep moving
  until E0 inside the conductor.
• This argument means that any excess charge on a
  conductor resides on the surface and that E is
  perpendicular to the surface of the conductor.

62
E-field due to charged conductor

• A sphere of radius a has a charge Q distributed
  uniformly throughout its volume. The sphere is
  surrounded by a conducting spherical shell of
  inner radius b and outer radius c. (See picture
  on the following slide). The conducting shell
  carries charge 5Q
• Find an expression for the charge density of the
  inner sphere.

63
E-field

• Find E for rlta, altrltb, bltrltc, rgtc
• What is the charge on the inner and outer surface
  of the conducting shell?

64
Example

• A solid, non-conducting sphere of radius R has a
  non-uniform volume charge density given by
•   where a is a constant.  Find E(r) in the range
  0 lt r lt R.

65
Spherical co-ordinates

• Volume element
• dVr2sinqdqdf

66
Cylindrical symmetry

• A long straight wire of radius a carries a charge
  per unit length l. What is the E field around the
  wire for rlta and for rgta?

67
Example

• A long, non-conducting cylindrical shell of
  inner radius a and outer radius b has a
  non-uniform charge density given by
• where ro is a constant.
• (a) Find the electric field in the region r lt a.
• (b) Find the electric field in the region a lt r lt
  b.
• (c) Find the electric field in the region b lt r.

68
Electrostatic potential

• The potential approach to electrostatics uses
  work and energy instead of force and
  acceleration. Compare this approach to the two
  techniques used in mechanics forces/accelerations
  etc versus potential and kinetic energy.
• Which approach is more fundamental?
• Why two approaches?

69
Definition of potential

• In any volume of space where there is an E-field
  we can define an electric potential between two
  points as the negative of the work done per unit
  charge by the E-field to move from one point to
  the other.

70
The point charge

• Where is a good choice for zero potential?
• What is the potential at a radius R?

71
Potential due to charge distributions

• Single point charge
• Charge distributions

72
Example

• Find the potential in the middle of the square of
  charges.

73
Example

• A line of charge of length L is situated along
  the x-axis as shown.  The line has a linear
  charge density given by l(x) ax, where a is a
  constant. What is the electrostatic potential at
  point P, a distance D from the end of the line?

74
Example

• A solid sphere of radius R has a total charge Q
  distributed uniformly throughout its volume. 
  What is the electrostatic potential at the center
  of the sphere (relative to infinity)?

75
Finding E from V

• The electric field at a point in space can be
  found from the gradient of the potential V.
• Gradient is given by the partial differentiation
• In words, the component of E in any direction is
  the rate at which V changes in that direction.

76
Surface of a conductor

• The surface of a conductor must be an
  equipotential. Why? If V changes then there is a
  gradient of V leading to an electric field E,
  leading to a force on the surface charges of the
  conductor.

77
Example

• V as a function of position in Cartesian
  coordinates is given by
• What is E at the point (2,-1,3)

78
Capacitance

• We have already defined capacitance earlier as
  the ratio of charge to voltage for a pair of
  conductors carrying opposite charges Q. CQ/V
• The voltage difference between conductors can be
  found by
• which means we can determine how C depends on
  conductor geometry.

79
Finding C of pairs of conductors

• Assume each conductor has equal magnitude charge
  but opposite sign.
• Find E (Gausss law usually) in the region
  between the conductors
• Find the potential difference between the
  conductors, DV, by integrating E.dl along a path
  joining the conductors
• Use CQ/DV to find an expression for C.

80
Example

• Find an expression for the capacitance of a pair
  of parallel conducting plates of area, A, and
  separation distance, d.
• How does this result guide you in determining how
  to combine capacitors in parallel?

81
Combining capacitors in series

• What quantity is the same for the capacitors in
  series? (Hint rhymes with barge.)
• What single capacitor CT could replace C1 and C2
  in series?

82
Example

• Find an expression for the capacitance of two
  concentric spheres of radii a and b (b gt a)
• What is the capacitance of an isolated metal
  sphere?

83
Energy in a capacitor

• The stored energy in a capacitor can easily be
  shown to be
• Energy density in a parallel plate capacitor

84
Dielectrics

• Insulating materials (also called dielectrics) in
  a capacitor alter the E field.
• In insulators the charges are not completely free
  to move but they can distort i.e. positive tends
  to move in one direction whereas the negative
  moves in the opposite. Polarization.

85
Free and bound charges

• Use Gausss law to find E in the dielectric
• What is q in term so k and q?

86
Example

• A coaxial cable consists of a wire of radius a
  surrounded by a metal cylinder of radius b. The
  intervening space is filled with a dielectric
  with k.
• What is the capacitance of 1 m of the coaxial
  cable?
• What is C if a2 mm, b5 mm k9?
• What is the free charge if V20 V?
• What is the bound charge?

87
Magnetic fields

• Electric fields are created by charges
• Magnetic fields are created by currents (i.e.
  moving charges)
• Electromagnets i.e. due to real currents in wires
• Spin and angular momentum of electrons in
  atomsmagnetic materials
• We define E due to the force it exerts on another
  charge. Same method for magnetic field.

88
Magnetism

• Magnetic field representationsimilar to E-field
  a vector field based on the force exerted on
  charges (but only moving charges).
• No magnetic monopolesi.e. no isolated sources of
  magnetic field

89
Force on a moving charge

• The magnetic field is given the symbol B.
• The force on a moving charge in a magnetic field
  B is given by
• Can a magnetic field accelerate a particle? (yes)
  Can it change a particles speed? (no).

90
Magnetic field, B

• Measured in unit of Teslas
• 1 T 1 Newton second/Coulomb meter 1
  Newton/Amp meter

91
Right Hand Rule
92
Example

• An electron moves with a velocity of 3 x105 m/s
  in the y direction in a region of magnetic field
  given by
• What is the magnetic force acting on the electron?

93
Example

• An ion of mass 3.2 x 10-26 kg and charge e is
  accelerated through a voltage of 833 V. The
  charge then enters a uniform magnetic field 0f
  0.92 T as shown in the figure. What is the radius
  of the resulting circular motion? What is the
  period and linear frequency?

94
Magnetic force on a wire

• Consider the force on each of the charges that
  makes up the current in a wire of length L and
  cross-sectional area A.
• Force on a wire

95
Example

• A conducting rod is supported horizontally by two
  conducting springs in a uniform horizontal
  magnetic field. The linear mass density of the
  rod is 0.040 kg/m, and the magnetic field
  strength is 3.6 T.  Find the current through the
  rod that results in zero tension in the support
  springs.

96
Force on a wire loop in a B field

• Operating basis of many electric motors
• N turns of wire in loop. What is the torque t?

97
Magnetic dipole moment

• A current loop behaves like a little bar magnet
  aligning with a magnetic field.
• The magnitude of the dipole moment is
• The direction of the dipole moment vector is
  given by right hand rule

98
Torque on a dipole

• From original definition
• Now with mNiA

99
Biot-Savart Law

• Moving charges are affected by magnetic fields
  similarly moving charges (currents) create
  magnetic fields.
• Biot-Savart law

100
Example

• A wire is configured as shown in the diagram.
  What is the magnetic field at P due to a current
  I 75 ma. a 1.2 cm b 3.5 cm, qp/3 radians.

101
Test 2 topics

• You need to be able to
• derive the E-field for planar, cylindrical, and
  spherical geometries using Gausss law (with and
  without dielectrics)
• Integrate E to get the voltage between conductors
  and hence find capacitance
• Find the potential, V, due to a collection of
  charges
• Find the force (magnitude direction) on charges
  in a magnetic field
• Determine the magnetic field due to a current
  using the Biot-Savart law (HW questions Ch 29
  6,24)

102
Amperes Law

• Amperes law is to magnetism what Gausss law is
  to electrostatics.
• This method works in cases with high symmetry
  where the properties of the B field can be
  inferred, figured out, whatever.

103
Current enclosed

• Current density J is the current per unit area
  through a wire.
• Example What is J for a 6 mm diameter wire
  carrying a uniformly distributed current of 5
  Amps?

104
Example

• Use the Biot-Savart law to determine the
  direction of the magnetic field around a current
  in a long-straight wire.
• Use Amperes law to find the magnitude of the
  magnetic field a distance r from the center of
  the wire.
• Find the value of the magnetic field magnitude a
  distance of 10 cm from a wire carrying a current
  of 1.0 A.

105
Example

• A current of 2.5 A flows through a solid wire
  cable of radius 2.5 cm.  Assuming that the
  current is distributed uniformly across the cross
  section of the wire, find the magnetic field
  magnitude inside the wire at a distance of
  one-half the wires radius from its central axis.

106
Example

• A solenoid consists of loops of wire carrying a
  current I. Let the solenoid be wrapped with n
  turns-per-unit-length of wire.  Find an
  expression for the magnetic field inside the
  solenoid.

107
Faradays law and Lenzs law

• Faradays law describes how magnetic fields can
  create voltages i.e. we are now connecting
  magnetic and electric phenomena
• In words a time varying magnetic flux through a
  circuit loop creates a voltage difference between
  the ends of the loop.
• Lenzs law indicates the polarity of the voltage

108
Faradays law

• Vind is the induced voltage between the ends of
  the loop, fM is the magnetic flux through the
  loop
• Flux through 1 loop is given by
• Flux through N loops is N fM

109
Lenzs law

• The voltage induced is always such as to keep the
  flux through the circuit constant. The direction
  of the voltage is oriented to create a current in
  the loop such that the flux remains the same.
• For example, what is the current direction in the
  loop below as B increases into the page?

110
Solving Faraday law problems

• Draw a picture of the configuration
• Calculate an expression for the flux through the
  circuit
• Determine the voltage across the loop from
• Determine the polarity of the voltage from Lenzs
  law.

111
Example

• A single loop of wire has a rectangular cross
  section of area A 8.0 x 104 m2 and contains a
  resistor of resistance R 2.0 W. The loop is in
  a region of uniform magnetic field that points
  perpendicularly into the coil from your
  perspective. The magnetic field magnitude
  decreases linearly from Bi 2.5 T to Bf 0.50 T
  over a time interval of 1.0 s. Find the
  magnitude and direction of the current that is
  induced in the resistor while the magnetic field
  is changing.

112
Example

• A small loop of 250 turns and radius r 6.0 cm
  is inside a large solenoid with 400 turns/m.  The
  axis of the small loop is coincident with the
  axis of the solenoid. The current in the solenoid
  windings varies with time according to
• where Io 30 A and a 1.61 s-1.  Find the
  voltage induced around the small loop at t 1.0
  s.

113
Motional EMF

• A metal rod moving through a uniform B field as
  shown. The motion creates a force on the charges
  in the rod that causes them to move in the rod.
  What is the voltage between the ends of the rod?
  (2 methods)

114
Example

• A metal rod of length L has one end a distance a
  from a long straight wire carrying a current I. 
  The rod has a velocity that is parallel to the
  current in the wire. Find the induced voltage
  across the rod and specify the polarity.

115
Faradays law big picture

• The fundamental point of Faradays law is that a
  time-varying magnetic flux, fM, leads to an
  induced voltage and thus an E-field.
• In the briefest of terms
• A changing magnetic field produces an electric
  field.

116
Inductance

• Earlier we defined capacitance as a ratio of
  stored charge to voltage for 2 conductors.
• Here we define a similar quantity, inductance,
  that relates magnetic flux through circuit, NfM,
  to the current, i.

117
Inductance of a solenoid

• A solenoid has a length L, cross-sectional area,
  A, and a total of N turns. What is the
  inductance?
• Note that inductance only depends on the
  geometrical properties of the object.

118
Inductors in circuits

• The definition of inductance says
• Faradays law says
• Thus, in terms of inductance
• Sign of VL depends on i increasing or decreasing

119
RL circuit

• When the switch is closed what does the circuit
  do? What is the direction of VL?

120
RL behavior

• With voltage source
• Without voltage source
• What do these equations look like graphically?

121
Energy stored in an inductor

• Consider the circuit below. By the loop rule
• Multiply both sides by i interpret each term
• Conclusion

122
Energy density in a B field

• Consider a solenoid. Inductance per unit length
  is Lmon2A
• Now use the energy stored in an inductor formula
  UB and rearrange to find the energy density uB
  (i.e. energy per unit volume).

123
Energy density in E and B fields

• Compare the energy density in E and B fields
• Remember energy density is equivalent to pressure.

124

• A circuit contains 2 resistors, a power source,
  inductor and switch as shown.  At t 0, the
  switch is thrown from neutral to A. Data  R1
  4.0 W   R2 7.0 W   L 8.0 mH  V 6.0 V
• (a) What is the time constant of this circuit?
• (b) What is the maximum current that can flow in
  this circuit?
• (c) What is the current in the circuit at to
  250 ms?
• (d) What is the rate at which energy is being
  stored in the magnetic field of the inductor
  coils at time to?
• (e) At time to the switch is set to B.  How much
  later has the rate at which energy loss due to
  Joule heating in the resistors is down by a
  factor of 10 from its value when the switch was
  set to position B?
•  

125
LC oscillations

• LC circuit Apply loop rule to get

126
Example

• Show that the total energy in an LC circuit (that
  is, the energy in the electric field between the
  capacitor plates and the magnetic energy in the
  magnetic field around the inductor coils) is a
  constant.

127
Example

• An LC circuit with inductor L80 mH and capacitor
  C5.0 mF. The initial charge on the capacitor and
  current through the inductor at t0 are Qo1.2 mC
  and Io 6.7 mA.    Find
• (a) the natural frequency of oscillation of the
  circuit.
• (b) the phase constant.
• (c) the charge amplitude.
• (d) the current amplitude.
• (e) the period of the current oscillations.
• (f) the charge on the capacitor plates when t
  2.0 ms.
• (g) the current through the inductor coils when t
  2.0 ms.
• (h) the time at which the energy stored in the
  magnetic field around the inductor coils first
  goes to zero after t 0.
• (i) the rate at which the energy being stored in
  the electric field between the capacitor plates
  is changing at t 2.0 ms.

128
Maxwells Equations

• Maxwell added one item to the effects we have
  studied so fara changing E field can act like a
  current and create a B field. Compare this with a
  changing B creating an E field (Faradays law).
• The final four

129
The wave equation

• Maxwell combined the 4 equations to show that in
  a charge free and current free region (Q0 and
  I0) the equations combined to predict a coupled
  electromagnetic wave. The E and B of this wave in
  vacuum are described by

130
Wave properties

• E and B are perpendicular and they are in phase
• The direction of the wave is described by
• The wave speed v (c in vacuum) is given by
• EcB

131
Poynting Vector

• The flow of energy in an EM wave is described by
  the Poynting vector S defined by
• We can show that the wave intensity, I, is given
  by

132
Energy density pressure

• Imagine an EM wave carrying an energy U in a time
  Dt onto an absorbing surface. What is the force
  on the surface?

133
Example

• A laser of average power output 10 mW produces a
  coherent beam of radius r 0.80 mm.
• (a) What is the average output intensity of the
  laser?
• (b) What is the average energy density in the
  beam?
• (c) What is the electric-field amplitude of the
  light wave produced by the laser?
• (d) What is the magnetic-field amplitude in the
  output beam?

134
Example

• The intensity of solar radiation at the earths
  position is I 1000 W/m2.  The radius of the
  earth is RE 6.37 x 106 m, the masses of the
  earth and sun are ME 5.98 x 1024 kg and MS
  1.99 x 1030 kg, the distance from the earth to
  the sun is D 1.496 x 1011 m, and the universal
  gravitational constant is G 6.67 x 1011 N.
  m2/kg2. 
• (a) What is the average power radiated by the
  sun?

135
Example (continued)

• The albedo of a planet is defined to be the
  ratio of the total light reflected from it to the
  total light incident on it.  The approximate
  albedo of Earth 0.34.
• (b) What approximate total force acts on the
  earth due to radiation pressure?
• (c) What is the ratio of the gravitational force
  exerted on the earth by the sun to the total
  force due to radiation pressure?

136
Introduction to Optics

• Optics is generally the study of a narrow part of
  the entire EM spectrumprimarily the part we can
  see (wavelengths from 700 nm to 400 nm)

137
Wave basics

• In optics the wavelength (rather than the
  frequency) is generally used to describe light of
  a fixed color.
• Of course wavelength and frequency are related by
• Red light 650 nm, green light 540 nm, blue 470 nm.

138
Geometrical Optics

• Light is a wave and waves diffract.
• Diffraction is negligible if the apertures and
  objects that the radiation interacts with are
  much larger than a wavelength.
• Geometrical optics describes light propagation
  where diffraction is not important.
• Light travels as rays. (Also called ray optics).

139
Index of refraction

• Light in vacuum travels at c (3 x 108 m/s)
• In transparent materials (dielectrics) the speed
  of light slows. The extent of slowing depends on
  the materials and it is described by the
  refractive index, n, of the material.
• Glass n1.5 (approximately) i.e. vglassc/n
• Water n1.33
• Picture of a wave in a materialwhat changes?

140
Chromatic dispersion

• The refractive index in a material changes with
  the wavelength of the radiationthis effect is
  called dispersion.
• Normal dispersion means that the index rises with
  decreasing wavelength.
• Dispersion causes a prism to split white light
  into its colors and causes different colors to
  focus at different distances from a lens.

141
Example

• Orange light of wavelength 600 nm in air enters
  water having an index of refraction of 1.33.
• (a) What is the frequency of this light?
• (b) What is the speed of the light in the water?
• (c) What is the wavelength of the light in the
  water?
• (d) If a person with normal vision were swimming
  under water and looked at this light,
  approximately what color would the person see?

142
Refraction

• When a ray of light moves from one material to
  another the change in velocity causes the ray to
  deviate. Snells law

143
Total-internal-reflection

• Consider a ray traveling from a high refractive
  index material into a low refractive index
  material. Above what angle of incidence will all
  of the light be reflected? Use the example of
  glass (ng1.5) and air (n1)

144
Example

• A small fish is swimming 1.2 m below the surface
  of a calm pond. You are standing on a small
  walking bridge over the pond looking directly
  down at the fish.  How far beneath the waters
  surface does the fish seem to be to you, given
  that the index of refraction of the pond water is
  1.35?

145
Example

• A narrow beam of white light is incident at an
  angle of 50o from the normal of a 60o prism made
  of fused quartz.   Find the angular dispersion of
  the light emerging from the far side of the
  prism.
• n(656nm)1.4564
• n(588nm)1.4585
• n(486nm)1.4631

146
Example

• A small dot is placed at the center of a piece of
  paper. Glass of thickness 1.26 cm and index of
  refraction 1.48 is placed on top of the paper. 
  What is the smallest radius of a coin that, when
  placed on top of the glass and centered over the
  dot, prevents anyone from being able to see the
  dot from above the glass?

147
Lenses and image formation

• Lenses are created by making one or both surfaces
  of a glass plates spherically curved.
• By applying Snells law to rays hitting the lens
  parallel rays of light can be brought to a focus.
  
• http//www.fhsu.edu/ktrantha/JavaOptics/javalens.
  html

148
Light from objects

• Light from illuminated objects diverges from
  every point of the object. Every point of the
  object is a source of scattered light rays
  traveling in all directions.
• Light rays from distant objects is essentially
  parallel. Thus, direct sunlight consists of
  parallel rays of light.

149
Lens fundamentals

• Convex Lensconverging lens

150
Lens Fundamentals

• Concave lensdiverging lens (note F1 and F2
  reversed)

151
Lenses and image formation

• A lens intercepts a selection of the scattered
  light from an object and focuses the rays. In
  order to determine the position of an image
  formed by a lens we consider only 3 easy to
  calculate rays.
• Ray 1 A ray leaving the tip of the object
  traveling parallel to the optical axis will pass
  through the focal point F2 after passing through
  the lens.
• Ray 2 A ray leaving the tip of the object and
  passing through the focal point F1 will emerge
  from the lens traveling parallel to the optical
  axis.
• Ray 3 The ray leaving the tip of the object and
  passing through the center of the lens will
  emerge from the lens undeviated.

152
Real and virtual images

• If the rays after the lens converge to a point
  then the image is real. Real means that it can be
  displayed on a screen placed at the point where
  the rays converge.
• If the rays after the lens diverge then the image
  is said to be virtual. A virtual image cannot be
  displayed on a screen but they can be observed by
  eye. The lens in the eye makes a real image on
  the retina.

153
The lens formula

• The positions of the image and object positions
  are given by
• f is the focal length (positive for converging
  lens negative for diverging lens)
• di is positive for a real image, negative for a
  virtual image
• do and di are positive on left and right of lens
  respectively

154
Magnification

• The magnification of the lens is given by how
  much bigger the image is compared to the object
• By examining the center ray it should be clear
  that

155
Example

• An object is placed 10 cm from a 15-cm focal
  length converging lens.
• (a) Find and describe the image analytically.
• (b) Find and describe the image using a ray
  diagram.

156
Example

• A diverging lens has a focal length of magnitude
  15.2 cm.  You view an object of height 3-cm
  through the lens when it is 10 cm from the center
  of the lens.  Locate and describe the image using
  analytical means, and then draw a rough ray
  diagram to help support your calculations and
  conclusions.

157
Mirrors

• The ray reflection from a mirror the angle of
  incidence is equal to the angle of reflection. qi
  qr

158
Example

• You are looking at your image in a plane mirror
  mounted flat on the wall 1 meter away directly in
  front of you. The mirror does not come all the
  way down to floor level yet you can just see your
  shoes in the image from the mirror. How high off
  the floor is the mirror (assume you are 1.6 m
  tall).

159
Spherical mirrors

• Curved mirrors (concave and convex) can form
  images just like lenses
• For a spherical mirror with radius of curvature
  10 cm, at what point is the focal length?

160
The 3 rays of happiness for mirrors

• Ray 1 The ray leaving the tip of the object
  traveling parallel to the optical axis will pass
  through the focal point after reflecting from the
  mirror.
• Ray 2 The ray leaving the tip of the object and
  passing through the focal point will reflect from
  the mirror traveling parallel to the  optical
  axis.
• Ray 3 The ray leaving the tip of the object and
  passing through the center point of the mirror
  will be reflected from the mirror undeflected.

161
Ray diagram
di and do are positive on the reflecting side of
the mirror.
162
Example

• A 1.00 cm-high object is placed 10.0 cm from a
  concave mirror whose radius of curvature is 30.0
  cm.
• (a) Draw a ray diagram to approximately locate
  and describe the image.
• (b) Locate and describe the image using
  analytical methods.

163
Multi-lens systems

• Apply the lens formula to the lens nearest the
  objectfind the image. Use that image as the
  object for the second lens to find the image of
  the compound lens system.
• The magnification of a two lens system is just
  the product of the magnifications of successive
  lenses
• mtotal m1 m2 m3

164
Example

• A compound microscope consists of two converging
  lenses.  The objective lens is a short
  focal-length lens that is closest to the object
  being studied.  The eyepiece is the lens through
  which the viewer looks to see the enlarged image
  of the object. The distance between the two
  lenses should be substantially greater than the
  sum of the two focal lengths of the two lenses,
  and the object is typically placed just outside
  the focal point of the objective lens.
• In this problem we will study the magnification
  of an ant using a compound microscope.  The
  objective lens has a focal length of 1.0 cm, and
  the eyepiece has a focal length of 2.5 cm.  The
  ant being studied is placed 1.1 cm in front of
  the objective lens, which is 13.4 cm from the
  eyepiece.  The ant is 0.2 mm wide.
• Locate and describe the image of the ant as
  viewed through this microscope.

165
Pinhole

• How does the pinhole allow imaging?

166
Wave nature of light

• EM radiation, including light, is a wave
  phenomenon.
• The wave nature of light (or any wave phenomenon)
  does not become apparent until the light
  interacts with openings, barrier, etc that are on
  the order of size of the wavelength of the
  radiation.
• Wave effects diffraction interference

167
Properties of waves

• Principle of superposition 2 parts
• Two or more waves will pass through one another
  without their direction or speed being changed.
  Waves dont collide off each other like particles
  do.
• Where two waves overlap the amplitude at a point
  in space is just the sum of the individual wave
  amplitudes at that point.
• The second point leads to wave interfere.

168
Interference

• When two single frequency waves combine the
  relative phase between them determines the extent
  of interference.
• Two extreme cases are when the waves are exactly
  in phaseamplitudes add leading to constructive
  interference.
• When waves are exactly out of phase the
  amplitudes canceldestructive interference.

169
Huygens Principle

• Huygens, developed a method of predicting the
  propagation of waves. Each point of a wave front
  becomes a secondary source of new waves. This
  method illustrates (but does not really explain)
  effects like diffraction.

170
Demos

• http//www.ngsir.netfirms.com/englishhtm/Diffracti
  on.htm
• http//physics.uwstout.edu/PhysApplets/a-city/phys
  engl/huygensengl.htm

171
Path length difference phase

• If two waves travel different length paths to
  between two points the path difference can be
  expressed as a phase difference.

172
Path phase

• If the path difference is Dx then the phase
  difference (for waves of wavelength l) is
• Constructive interference occurs if Dx is 0, l,
  2l ie Df0, 2p, 4p ..
• Destructive interference occurs if Dx is l/2,
  3l/2, 5l/2 ie Dfp, 3p, 5p ..

173
Youngs double slit

• Demonstrated light was a wave
• Bright fringe
• Dark fringe

174
Intensity of the two slit diffraction

• Phasor concept
• At any point on the screen the time varying
  E-field from the two slits is given by
• The phase difference f is found from the path
  difference at angle q.

175
Phasor sum

• Individual phasors (left) and summed phasors
  (right)

176
Intensity

• The intensity of light is given by the square of
  the total E-field at any point on the screen
• The E-field varies across the screen because of
  interference
• Phasor picture and a little bit of mathematical
  jiggery-pokery

177
Example

• Light of wavelength 587.5 nm is incident normally
  on a set of two slits separated by a distance of
  0.20 mm, and then projected onto a distant
  screen.  The second-order maximum of the
  interference pattern thus produced is found to be
  at the position y2 2.0 mm.  Find the distance
  from the slits to the screen.

178
Single slit diffraction pattern

• Derivation of the single slit diffraction
  equation
• Minima given by

179
Intensity for single slit

• Phasors, but with a whole bunch of sources side
  by side across the opening.
• EM is the sum of all phasors end to end
• Eq is the amplitude we seek.
• I(q)Eq2

180
Phase, f

• The phase, f, in the diagram is the difference in
  phase between the paths to the screen from
  opposite ends of the slit opening.
• Path difference a sinq
• Phase difference

181
Intensity for a single slit

• Final result
• This is the sinc function. What is its value for
  a0?

182
Double slit / slit width
183
Double slit
184
Full double slit formula

• Double slit with slit width, a, and slit
  separation, d.

185
Example

• Light from a green laser (540 nm) shines through
  a set of double slits and is incident on the
  board at the front of the room.
• (a) By inspection of the pattern of laser light
  formed on the board, but without performing any
  distance measurements, estimate the ratio of the
  slit separation to the slit width.
• (b) Use distance measurements to help estimate
  the separation of the slits and the width of each
  slit.

186
Diffraction from a Grating
187
Gratings

• Grating are usually specified by the number of
  rulings or grooves per unit length. For example,
  300 lines per mm. You can get d easily from this
  definition.
• Diffraction orders are labeled m0, 1, 2
• Grating diffraction leads to much narrower
  well-defined bright diffraction spots because of
  the larger illuminated aperture.
• Resolution

188
Example

• Light from a green laser (540 nm) shines through
  a diffraction grating of width 4.5 cm and is
  incident on the board at the front of the room. 
• (a) What is the separation between adjacent
  grooves?
• (b) What is the linear groove density in the
  grating?
• (c) What is the total number of grooves in the
  grating?

189
Thin film interference

• An everyday example of thin film interference is
  the colorful pattern created by gasoline films on
  water in parking lot puddles.
• The effect is due to interference between light
  reflected from the top and bottom surfaces of the
  thin gas layer. Cancellation of light occurs when
  the phase difference between the 2 reflected rays
  differs by p, 3p, 5p
• The effect also occurs in any situation where
  there are two closely spaced reflecting surfaces.

190
Phase change on reflection

• Whenever light hits an interface between 2 media
  with different refractive indices some light is
  reflected and some transmitted.
• If the light is reflected from a low-index to
  high-index transition then the reflected light is
  inverted on reflection (i.e. it suffers a phase
  change of p radians).
• Zero phase changes occurs for a high n to low n
  transition.
• No phase change for transmitted light.

191
Path difference

• In a thin film interference problems
• figure out the path difference between the two
  rays and
• account for phase changes on reflection, if any.
• If the total phase change is 0, 2p, 3p.. the
  interference is constructive (bright fringe). If
  total phase is p, 3p/2, 5p/2 the phase change
  is destructive (dark fringe).

192
Example

• Thin films of material are often used as
  non-reflective coatings on camera lenses. A
  thin film of magnesium fluoride having an index
  of refraction of 1.38 coats a glass lens having
  an index of refraction of 1.55.  The film has a
  thickness of 0.73 mm. 
• What wavelengths in the visible region of the
  electromagnetic spectrum can be clearly seen upon
  reflection from the thin film?

193
Final Exam

• 7 problems
• Gausss law
• Circuits, R, RL, RC
• Mirrors/lenses and imaging
• Covers everything up to thin films but not
  polarization

194
Example

• One glass plate is positioned on top of another
  with the end of a hair between the plates at one
  end. The plates are illuminated from above with
  light of wavelength 589.0 nm from a sodium vapor
  lamp. An interference pattern is observed due to
  the variations in the thickness of the air layer
  between the glass plates. Use the interference
  pattern to estimate the diameter of the hair.

195
Polarization

• Polarization refers to the orientation of the
  E-field vector in an electromagnetic wave.
• Most light is unpolarized, i.e. the E-field of
  the light is equally distributed over all angles.
• A polarizer is a filter that allows one direction
  of the E-field pass through.

196
Polarizer

• E-field vector component parallel to the
  polarizer direction is allowed through.

197
Intensity

• We measure intensity not E-field by eye and using
  light meters.
• What intensity of unpolarized light is
  transmitted through a polarizer? What is its
  polarization angle after transmission?
• What is the total incident intensity?
• What intensity is transmitted?

198
Polarized light transmission

• If linearly-polarized light is incident on a
  polarizer what is the intensity of transmission
  as a function of angle, q, between the E-field
  and the polarizer?
• Law of Malus

199
Example

• If unpolarized light of intensity 30 mW/m2 falls
  on a horizontally oriented polarizer, what
  intensity is transmitted and what is the
  orientation of the transmitted lights E-field?
• If a second polarizer is added at 45 degrees to
  the horizontal, what intensity is transmitted
  through the combination and what is the
  orientation of the E-field?

200
Polarization on reflection

• Brewsters angle red p-pol. blue s-pol.

201
Brewsters angle

• The angle of minimum reflection of p-polarized
  light is given by

202
Example

• Discuss with as much detail as possible the
  Polaroid sunglasses commercial on TV in which
  swimmers beneath the waters surface in a
  swimming pool cannot be seen due to glare from
  the suns light reflected off of the water
  surface, but when Polaroid sunglasses are placed
  over the TV-camera lens, the swimmers can be seen
  clearly.