Download PHYS 241 Exam Review

PHYS 221 Exam 1 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
• Begin by drawing free body diagrams
• “Play” around with the problem
• Take fifteen to twenty minutes before the
exam to relax… no studying.
• Look for symmetries
Concepts
• Electricity
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Electrostatics
Coulomb’s Law
Principle of Superposition
Electric Field
Conductors vs. Insulators
Flux
Gauss’s Law
Potential Energy
Potential
Capacitance
Concepts
• Circuits
– Current
– Resistance/Resistivity
– Kirchoff’s Rules
• Magnetism
– Magnetic Fields
Electrostatics
• It may not have been explicit at this point, but
we have been operating under some
assumptions
• We have assumed that all of our charges are
either stationary or in a state of dynamic
equilibrium
• We do this because it simplifies the electric
fields we are dealing with and eliminates the
presence of magnetic fields
Coulomb’s Law
• What does it tell me?
– It tells you the force between two charged
particles
• Why do I care?
– Forces describe the acceleration a body undergoes
– The actual path the body takes in time can be
found from the acceleration using kinematics if
the acceleration is uniform
Coulomb’s Law
• Forces have magnitude and direction so
Coulomb’s law tells you both of these
– Magnitude: 𝐹 = 𝑘
𝑞1 𝑞2
𝑟12
2
– Direction: Along the line connecting the two
bodies. It is repulsive in the case of like charges,
attractive for opposite charges
Principle of Superposition
• What does it tell me?
– The electric force between two bodies only depends
on the information about those two bodies
• Why do I care?
– Essentially, all other charges can be ignored, the result
obtained in pieces and then summed… this is much
simpler
𝑛
𝐹𝑖 = 𝐹1 + 𝐹2 + ⋯ + 𝐹𝑛
𝑖=1
Electric Field
• What does it tell me?
– The force a positive test charge q would experience at
a point in space
𝐹
Universal
𝐸 ≡ lim+ ⇒ 𝐹 = 𝑞𝐸
𝑞→0 𝑞
• Why do I care?
– Calculating the force a particular charge feels doesn’t
directly tell you how other charges would behave
– The electric field gives you a solution that applies to
any charge, so it reduces your work
Electric Field
• Electric field at a point 𝑟 due to a point charge
at 𝑟′ with charge q
𝑞
Situational
𝐸 𝑟 =𝑘
𝑟 − 𝑟′
3
𝑟 − 𝑟′
k: Coulomb’s Constant
• Principle of superposition still applies
– You can sum individual fields due to discrete
charges
Conductors vs Insulators
• Conductors
– All charge resides on the surface, spread out to
reduce the energy of the configuration
– The electric field inside is zero
– The potential on a conductor is constant (i.e. the
conductor is an equipotential)
– The electric field near the surface is perpendicular
to the surface
Note: These are all logically equivalent statements,
but only apply in the electrostatic approximation
Conductors vs Insulators
• Insulators
– Charge may reside anywhere within the volume or
on the surface and it will not move
– Electric fields are often non-zero inside so the
potential is changing throughout
– Electric fields can make any angle with the surface
Flux
• Flux, from the Latin word for “flow,” quantifies
the amount of a substance that flows through a
surface each second
• It makes sense that we could use the velocity of
the substance at each point to calculate the flow
• Obviously we only want the part of the vector
normal to the surface, 𝑣𝑛 , to contribute because
the parallel portion is flowing “along” the surface
• Intuitively then we expect the flux to then be
proportional to both the area of the surface and
the magnitude of 𝑣𝑛
Flux
• For the case of a flat surface and uniform
electric field, it looks like this:
Gauss’s Law
• What does it tell me?
– The electric flux (flow) through a closed surface is
proportional to the enclosed charge
• Why do I care?
– You can use this to determine the magnitude of
the electric field in highly symmetric instances
– Flux through a closed surface and enclosed charge
are easily exchanged
3 Considerations for Gaussian Surfaces
Gauss’s law is true for any imaginary, closed surface and any
charge distribution no matter how bizarre. It may not be
useful, however.
1. The point you are evaluating the electric field at needs to
be on your surface
2. Choose a surface that cuts perpendicularly to the electric
field (i.e. an equipotential surface)
3. Choose a surface where the field is constant on the
surface
*Note this requires an idea of what the field should look like
Common Gauss’s Law Pitfalls
• Your surface must be closed
• The charge you use in the formula is the
charge enclosed by your surface
• The Gaussian surface need not be a physical
surface
• Start from the definition of flux and simplify
only if your surface/field allows it
𝑞𝑒𝑛𝑐
Universal
Φ𝐸 =
𝜀𝑜
Potential Energy
• In a closed system with no dissipative forces
Δ𝑃𝐸𝑒𝑙𝑒𝑐 + 𝑊 = 0
Situational
• The work done is due to the electric force so
𝑊 = 𝐹 ⋅ ∆𝑟 = 𝑞𝐸 ⋅ ∆𝑟 Situational:
Uniform electric
field and straight path
WARNING: Since charge can be negative, 𝐸 and 𝐹 might point
in opposite directions (this is called antiparallel) which would
change the sign of W
• This can be combined with the work-energy
theorem to obtain the velocity a charged particle
has after moving through an electric field
Potential
• What does it tell me?
– The change in potential energy per unit charge an
object has when moved between two points
Δ𝑃𝐸𝑒𝑙𝑒𝑐
Δ𝑉 ≡
= −𝐸 ⋅ ∆𝑟 Situational: Electrostatics
𝑞
• Why do I care?
– The energy in a system is preserved unless there is
some kind of dissipative force
– So the potential allows you to use all the conservation
of energy tools from previous courses (i.e. quick path
to getting the velocity of a particle after it has moved
through a potential difference)
Potential
• Word of caution:
– Potential is not the same as potential energy, but they are
intimately related
– Electrostatic potential energy is not the same as potential
energy of a particle. The former is the work to construct
the entire configuration, while the later is the work
required to bring that one particle in from infinity
– There is no physical meaning to a potential, only difference
in potential matter. This means that you can assign any
point as a reference point for the potential
– The potential must be continuous
Analogies with Gravity
• Electricity and magnetism can feel very abstract because we
don’t usually recognize how much we interact with these forces
• There are many similarities between gravitational and electric
forces
• The major difference is that the electric force can be repulsive
• Gravity even has a version of Gauss’s law
Charge
Force
Field
PE
Electricity
q
𝑄𝑞
𝑘 2𝑟
𝑟
𝐸=𝑘
𝑄
𝐹
𝑟
=
𝑟2
𝑞
𝑞Δ𝑉
Gravity
m
𝐺
𝑀𝑚
𝑟
𝑟2
𝑔=𝐺
𝑀
𝐹
𝑟
=
𝑟2
𝑚
𝑚 𝑔Δ𝑦
Capacitance
• What does it tell me?
– The charge that accumulates on two conductors is
proportional to the voltage between them
𝑄 = 𝐶∆𝑉
Q: charge on the capacitor’s plates, C: capacitance,
ΔV: potential difference across the capacitor
• Why do I care?
– Capacitors are vital components in electronics
– They can be used to temporarily store charge and energy,
and play an even more important role when we move to
alternating current systems
– Camera flashes, touch screen devices, modern keyboards
all exploit capacitance
Capacitance
• What does capacitance depend on?
– Geometry of the plates
– Material between the plates
– For parallel plates: 𝐶 =
𝜀𝐴
𝑑
C: capacitance, ε: permittivity of the material between the plates, A: area of
the plates (may or may not be square), d: distance between the plates
• Unit of capacitance is the Farad
– To demystify this, units are (meters*permittivity)
Capacitance
• Dielectric
– Put simply, a dielectric is a material (an insulator) that weakens the
electric field around it
– This allows more charge to be placed on the plates for the same
voltage (i.e. capacitance is increased)
– The permittivity of a dielectric tells you how it affects the capacitance
– The ratio of the permittivity of a dielectric and the permittivity of free
space is the dielectric constant
𝜀
Situational: Assumes steady fields
𝜅≡
𝜀𝑜
κ: dielectric constant, ε: permittivity of a material, εo : permittivity of free space,
𝐶𝑑
𝜅=
𝐶𝑣𝑎𝑐
Situational:
Assumes uniform
dielectric
Cd : capacitance with a uniform dielectric, Cvac: capacitance in the vacuum
Capacitance
• The permittivity of free space has no physical
meaning
• It merely changes physical quantities into their
appropriate SI units
Physical Units
SI Units
Length
Farads
Length/Charge
Volts
Length^2/Charge^2
Newtons
Length/Charge^2
Joules
Practice Problem
Practice Problem
Practice Problem
Practice Problem
Practice Problem
Quiz Questions
Quiz Questions
Quiz Question
Quiz Question