Physics 1502: Lecture 25 Today

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Physics 1502 Lecture 25Todays Agenda

• Announcements
• Midterm 2 NOT Nov. 6
• Following week
• Homework 07 due Friday next week
• AC current
• Resonances
• Electromagnetic Waves
• Maxwells Equations - Revised
• Energy and Momentum in Waves

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AC Current
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PhasorsLCR
Þ
ß
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PhasorsTips

• This phasor diagram was drawn as a snapshot of
  time t0 with the voltages being given as the
  projections along the y-axis.

• Sometimes, in working problems, it is easier
  to draw the diagram at a time when the current is
  along the x-axis (when i0).

From this diagram, we can also create a triangle
which allows us to calculate the impedance Z
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Resonance

• The current in an LCR circuit depends on the
  values of the elements and on the driving
  frequency through the relation

Suppose you plot the current versus w, the
source voltage frequency, you would get
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Power and Resonance in RLC

• Power, as well as current, peaks at w w 0.
  The sharpness of the resonance depends on the
  values of the components.
• Recall

• Therefore,

We can write this in the following manner (which
we wont try to prove)
introducing the curious factors Q and x
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The Q factor
Q also determines the sharpness of the resonance
peaks in a graph of Power delivered by the source
versus frequency.
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Lecture 25, ACT 1

• Consider the two circuits shown where CII 2 CI.
  
• What is the relation between the quality factors,
  QI and QII , of the two circuits?

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Lecture 25, ACT 2

• Consider the two circuits shown where CII 2 CI
  and LII ½ LI.
• Which circuit has the narrowest width of the
  resonance peak?

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Power Transmission

• How do we transport power from power stations to
  homes?
• At home, the AC voltage obtained from outlets in
  this country is 120V at 60Hz.
• Transmission of power is typically at very high
  voltages ( eg 500 kV) (a high tension
  line)
• Transformers are used to raise the voltage for
  transmission and lower the voltage for use.
  Well describe these next.
• But why?
• Calculate ohmic losses in the transmission lines
• Define efficiency of transmission

• Note for fixed input power and line resistance,
  the inefficiency µ 1/V2

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Transformers

• AC voltages can be stepped up or stepped down
  by the use of transformers.

iron

• The AC current in the primary circuit creates a
  time-varying magnetic field in the iron

V1
e
V2

• This induces an emf on the secondary windings due
  to the mutual inductance of the two sets of
  coils.

(secondary)
(primary)

• The iron is used to maximize the mutual
  inductance. We assume that the entire flux
  produced by each turn of the primary is trapped
  in the iron.

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Ideal Transformers (no load)

• The primary circuit is just an AC voltage source
  in series with an inductor. The change in flux
  produced in each turn is given by

• The change in flux per turn in the secondary
  coil is the same as the change in flux per turn
  in the primary coil (ideal case). The induced
  voltage appearing across the secondary coil is
  given by

• Therefore,
• N2 gt N1 Þ secondary V2 is larger than primary V1
  (step-up)
• N1 gt N2 Þ secondary V2 is smaller than primary
  V1 (step-down)
• Note no load means no current in secondary.
  The primary current, termed the magnetizing
  current is small!

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Ideal Transformers

• What happens when we connect a resistive load to
  the secondary coil?
• Flux produced by primary coil induces an emf in
  secondary

• emf in secondary produces current i2

• This current produces a flux in the secondary
  coil µ N2i2, which opposes the original flux --
  Lenzs law

• This changing flux appears in the primary circuit
  as well the sense of it is to reduce the emf in
  the primary...
• However, V1 is a voltage source.
• Therefore, there must be an increased current i1
  (supplied by the voltage source) in the primary
  which produces a flux µ N1i1 which exactly
  cancels the flux produced by i2.

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Transformers with a Load

• With a resistive load in the secondary, the
  primary current is given by

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Lecture 25, ACT 3

• The primary coil of an ideal transformer is
  connected to a battery (V1 12V) as shown. The
  secondary winding has a load of 2 W. There are 50
  turns in the primary and 200 turns in the
  secondary.

R

• What is the current in the secondcary ?

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Lecture 25, ACT 4

• The primary coil of an ideal transformer is
  connected to the wall (V1 120V) as shown. There
  are 50 turns in the primary and 200 turns in the
  secondary.

• If 960 W are dissipated in the resistor R, what
  is the current in the primary ?

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Fields from Circuits?

• We have been focusing on what happens within the
  circuits we have been studying (eg currents,
  voltages, etc.)
• Whats happening outside the circuits??
• We know that
• charges create electric fields and
• moving charges (currents) create magnetic fields.
  
• Can we detect these fields?
• Demos
• We saw a bulb connected to a loop glow when the
  loop came near a solenoidal magnet.
• Light spreads out and makes interference
  patterns.
• Do we understand this?

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f
(
x
f
(
)
x
x
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Maxwells Equations

• These equations describe all of Electricity and
  Magnetism.
• They are consistent with modern ideas such as
  relativity.
• They describe light !

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Maxwells Equations - Revised

• In free space, outside the wires of a circuit,
  Maxwells equations reduce to the following.

• These can be solved (see notes) to give the
  following differential equations for E and B.

• These are wave equations. Just like for waves on
  a string. But here the field is changing instead
  of the displacement of the string.

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Plane Wave Derivation

• Step 1 Assume we have a plane wave propagating
  in z (ie E, B not functions of x or y)

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Plane Wave Derivation

• Step 4 Combine results from steps 2 and 3 to
  eliminate By

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Plane Wave Derivation

• We derived the wave eqn for Ex

• How are Ex and By related in phase and magnitude?

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Review of Waves from last semester

• The one-dimensional wave equation

• A specific solution for harmonic waves traveling
  in the x direction is

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E B in Electromagnetic Wave

• Plane Harmonic Wave

where
Nothing special about (Ey,Bz) eg could have
(Ey,-Bx)
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Lecture 25, ACT 5

• Suppose the electric field in an e-m wave is
  given by
• In what direction is this wave traveling ?

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Lecture 25, ACT 5

• Suppose the electric field in an e-m wave is
  given by

• Which of the following expressions describes the
  magnetic field associated with this wave?

(a) Bx -(Eo/c)cos(kz wt)
(b) Bx (Eo/c)cos(kz - wt)
(c) Bx (Eo/c)sin(kz - wt)
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Velocity of Electromagnetic Waves

• The wave equation for Ex (derived from
  Maxwells Eqn)

• Therefore, we now know the velocity of
  electromagnetic waves in free space

• Putting in the measured values for m0 e0, we
  get

• This value is identical to the measured speed of
  light!
• We identify light as an electromagnetic wave.

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The EM Spectrum

• These EM waves can take on any wavelength from
  angstroms to miles (and beyond).
• We give these waves different names depending on
  the wavelength.

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Energy in EM Waves / review

• Electromagnetic waves contain energy which is
  stored in E and B fields

• Therefore, the total energy density in an e-m
  wave u, where

• The Intensity of a wave is defined as the
  average power transmitted per unit area average
  energy density times wave velocity