Download PHYS 241 Final Exam Review

PHYS 241 Final Exam Review
Kevin Ralphs
Overview
• General Exam Strategies
• Concepts
• Practice Problems
General Exam Strategies
• Don’t panic!!!
• If you are stuck, move on to a different
problem to build confidence and momentum
• “Play” around with the problem
• Take fifteen to twenty minutes before the
exam to relax… no studying.
• Dimensional analysis is a good tool, but can
give false results
Concepts
•
•
•
•
EMF
Faraday’s Law
Inductance
AC Circuits
–
–
–
–
RMS
Reactance
Impedance
Phasors
• Displacement Current
• Electromagnetic Waves (Light)
– Wave/Particle Duality
– Poynting Vector
Concepts
• Optics
– Refraction
• Index of Refraction
• Snell’s Law
– Total Internal Reflection
– Malus’s Law
– Mirrors
– Lenses
– Diffraction
Electromotive Force (EMF)
• What does it tell me?
– The change in potential energy per unit charge an object
has when moved along a path
Δ𝑈
ℰ≡
𝑞
– It can also refer to the voltage measured across two
terminals
• Why do I care?
– So far we have considered conservative electric fields
which have scalar potentials
– For non-conservative fields, the change in potential energy
becomes path dependent and EMF is accounting for that
Electromotive Force (EMF)
• Why do I care?
– If a particle is free to move around in space, this is
not all that helpful, but when they are constrained
to move on a specified path (like an electronic
circuit), it becomes well-defined.
Note:
1. This is not a force, it has units of volts
2. This is not a potential, the path taken matters
very much
Motional EMF
• When a conductor moves through a magnetic
field, it acquires an EMF (this is more along
the lines of the two terminal definition)
• This happens because a Lorentz force from the
magnetic field shuffles charges to opposite
ends of the conductor
• This sets up a voltage like a parallel plate
capacitor bringing the charges into an
equilibrium
Motional EMF
Farraday’s Law
• Two earlier approximation schemes
– Electrostatics
• Stationary charges
• Conducting charges at equilibrium
𝐸 ⋅ 𝑑𝑙 = 0
𝛻 × 𝐸=0
– Magnetostatics
• Steady Currents
Farraday’s Law
• In electrodynamics we allow single charges to
move
• This causes time varying magnetic fields
bringing Farraday’s law into effect
𝜕𝐵
𝛻 × 𝐸=−
𝜕𝑡
𝑑
𝐸 ⋅ 𝑑𝑙 = −
𝐵 ⋅ 𝑛𝑑𝐴 = Ɛ
𝑑𝑡
Farraday’s Law
• What does it tell me?
– A changing magnetic field creates a non-conservative
electric field
– Anything that affects that flux integral induces an EMF
in a loop
• Why should I care?
– Without this law, you could not see, there would be
no cell phones or radio: electromagnetic waves exist
because of this
– Inductors and transformers exploit this phenomenon
Lenz’s Law
• What does it tell me?
– When the flux through a loop changes, a current is
produced that fights this change
• Why should I care?
– This principle is how you determine the direction
of an induced current
Lenz’s Law
• If you are having problems with this, you are not alone
– People spend thousands of hours researching this (no
kidding)
• The idea is to find the direction of the induced
magnetic field and use the right hand rule to find the
current
• To find the direction of the induced field
– Note the direction of the original field through the loop
– Determine whether this field is getting stronger or weaker
– The direction of the induced field will maintain the status
quo
Inductance
• What does it tell me?
– The flux through a loop is proportional to the
currents on conductors in the vicinity (including
itself)
Φ𝑀 = 𝐿𝐼 +
𝑀𝑖 𝐼𝑖
– This is a direct consequence of the principle of
superposition and magnetic fields being
proportional to the currents that create them
Inductance
• Why should I care?
– This is the sister component to the capacitor making it one
of the most fundamental electronic components
Capacitor
Inductor
Depends on geometry and
material between the plates
Depends on geometry and
material in intervening space
Proportionality between charge
and voltage
Proportionality between flux and
current
Stores energy in an electric field
Stores energy in a magnetic field
Causes current to lag voltage
Causes current to lead voltage
𝑄
𝐶
Current starts at maximum and
drops to zero
𝑑𝐼
𝑑𝑡
Current starts at zero and
increases to maximum
𝑉=
𝑉=𝐿
Alternating Current (AC) - RMS
• What does it tell me?
– RMS is a type of averaging
– First square the wave form, then we average and
take the square root
• Why should I care?
– This allows us to keep a form of the Joule heating
law
𝑃𝑎𝑣𝑔 = 𝐼𝑟𝑚𝑠 2 𝑅
AC - Reactance
• What does it tell me?
– Capacitors and inductors resist changes in the state of the
circuit – Reactance is a measure of this
• Why should I care?
– Calculating the voltages on capacitors and inductors in an
AC circuit can be complicated
– Reactance give you a direct link between the average
voltage across these components and the RMS current in
an Ohm’s law type format
– It also shows how the frequency of the applied voltage
affects the system
1
𝑋𝐿 = 𝜔 𝐿 𝑋𝐶 =
𝜔𝐶
AC - Impedance
• What does it tell me?
– It represents the relationship (magnitude and
phase difference) between the applied voltage
and the current
• Why should I care?
– Impedance provides a compact way to carry a lot
of information about your circuit
AC - Impedance
• Since the impedance carries phase information, it
is a complex number
1
𝑖
𝑋𝐿 = 𝑖 𝜔 𝐿 𝑋𝐶 =
=−
𝑖𝜔𝐶
𝜔𝐶
1
𝑍 = 𝑅 + 𝑋𝐿 + 𝑋𝐶 = 𝑅 + 𝑖(𝜔𝐿 −
)
𝜔𝐶
• The circuit is at resonance when the impedance is
a real number
– This corresponds to maximum power transfer to the
resistors
AC - Phasors
• A phasor is a graphical representation of the
relationship between voltage and current in a
system
• This exploits the power of complex numbers
as both vectors and rotations
• The phasor rotates through the complex plane
and the real projections of the phasor give the
measured value
• See Demonstration
Displacement Current
• What does it tell me?
– A changing electric field produces a magnetic field
as if there was a current flowing that is
proportional to the change in flux
𝜕𝐸
𝛻 × 𝐵 = 𝜇𝑜 𝐽 + 𝜀𝑜
𝜕𝑡
𝑑Φ𝑒
𝐵 ⋅ 𝑑 𝑙 = 𝜇𝑜 𝐼 + 𝐼𝐷 = 𝜇𝑜 𝐼 + 𝜀𝑜
𝑑𝑡
Displacement Current
• Why do I care?
– The correction completes Ampere’s law bringing it
in agreement with the Biot-Savart Law
– Like Faraday’s law, this allows for the propagation
of electromagnetic waves
Poynting Vector
• What does it tell me?
– Energy and momentum can be carried away by
electromagnetic waves
1
𝑆=
𝐸 ×𝐵
𝜇𝑜
𝜕
−
𝑢𝑑𝑉 =
𝑆 ⋅ 𝑛𝑑𝐴 +
𝐽 ⋅ 𝐸𝑑𝑉
𝜕𝑡 𝑉
𝜕𝑉
𝑉
Change in internal
energy
Energy flowing out
Work done inside
Poynting Vector
• Why do I care?
–
–
–
–
It is a conservation law
Newton’s third law fails without it
Hints at the need for special relativity
The intensity (power) of light is defined as the time
average of the vectors magnitude
𝐼≡ 𝑆
– Radiation pressure is related to the intensity
𝐼
𝑃𝑅𝑎𝑑 ≡
𝑐
Index of Refraction
• What does it tell me?
– The ratio of the speed of a wave in a reference medium
(we choose the vacuum) and another medium
𝑐
1
𝑛≡ =
𝑣
𝜀𝑟 𝜇𝑟
• Why should I care?
– The index of refraction influences nearly all optical
phenomena in some way
• Depends on electrical and magnetic properties of the
medium – sensitive to frequency (i.e. 𝑛: 𝜔 ⟼ ℝ)
Snell’s Law
• What does it tell me?
– The relationship between
the indices of refraction and
the angles of refraction and
reflection
𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2
• Why should I care?
– This concept is the “building block” for more
advanced concepts such as thin film diffraction
• Remember that ALL angles are measured from
the NORMAL of the surface
Malus’s Law
• What does it tell me?
– How the intensity of polarized light is affected by a
polarizer
𝐼 = 𝐼𝑜 cos2 𝜃
– Your book defines theta to be the angle between
the transmission axes of two polarizers
– Alternatively, it is the angle between the plane of
polarization and the transmission axis of the
polarizer
– If the light is unpolarized, the intensity if halved
Assumptions/Conventions
• Wavelength of light is much shorter than the
length scale of the geometry
– Treat light as rays; i.e. no bending
• Small angle deviations from the optical axis
– Spherical surface is nearly parabolic
sin 𝜃 ≈ 𝜃
• The biggest challenge in applying the formulae
is following the correct sign convention
Mirrors
Rules for Ray Diagrams
Parallel Rays
Reflected through focal point
Focal Rays
Reflected parallel to optical axis
Radial Rays
Reflected back on itself
Sign Convention
s is positive if object is on the incident-light side
s’ is positive if the image is on the reflected-light side
R is positive if the mirror is concave
Lenses
Rules for Ray Diagrams
Parallel Rays
Refracted through focal point
Focal Rays
Refracted parallel to optical axis
Central Rays
No deflection when refracted
Sign Convention
s is positive if object is on the incident-light side
s’ is positive if the image is on the refracted-light side
r is positive if center of curvature is on the refracted-light side
Diffraction
• What does it tell me?
– How a wave behaves near objects
– Only an appreciable affect when the length scale
of the wave and the geometry are similar
• Visible light: 400nm – 700nm
• Sound waves: 17mm – 17m
Interference
• What does it tell me?
– How waves mix together
– Based on the principal of superposition
𝐴 cos 2𝜋𝑓1 + 𝐴 cos 2𝜋𝑓2 = 2𝐴 cos 2𝜋
𝑓1 − 𝑓2
𝑓1 + 𝑓2
cos 2𝜋
2
2
– Always occurs, but is especially noticeable when the waves
are coherent
Interference
• There are two main sources of interference
that we will consider
– Path length difference
2𝜋𝑑
∆𝜑 =
𝜆
– Reflected waves can pick up a phase shift when
going into a medium with a higher index of
refraction
𝑛1 − 𝑛2
𝐴𝑅 = 𝐴𝐼
𝑛1 + 𝑛2
Main Strategy
• For any kind of diffraction, the game is always
about counting up phase shifts; these can be
expressed in terms of angles or wavelengths
– Angles
• Constructive Interference: Even multiples of π
• Destructive Interference: Odd multiples of π
– Wavelengths
• Constructive Interference: Integer multiples of λ
• Destructive Interference: Odd half-integer multiples of λ
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