Download PHYS 241 Exam Review

PHYS 241 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
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Electrostatics
Coulomb’s Law
Principle of Superposition
Electric Field
Continuous Charge Distributions
Conductors vs. Insulators
Gauss’s Law
Potential
Capacitance
Electrostatics
• Our study of electric fields so far has been based
on a few assumptions
• These assumptions collectively are known as the
electrostatic approximation
• Basically we assume that our systems have to
come to a dynamic equilibrium before we do our
calculations
• We will be ignoring transitory behavior or steady
state behaviors (no currents or 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 in two ways
1. Use integration to get the particle’s velocity as a function
of time, then integrate again to gets its position
2. Kinematic equations (the result when method 1. is applied
in the case of constant acceleration)
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?
– A vector proportional to the force a positive test
charge would experience at a point in space
• 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 due to a point charge at distance
r with charge q
𝑞
𝐸 = 𝑘 2𝑟
𝑟
• Principle of superposition still applies
– You can sum individual fields due to discrete
charges
– You can integrate continuous charge distributions
where the charge becomes 𝑑𝑞 and the field
becomes 𝑑𝐸
Continuous Charge Distributions
• Motivation for the equation:
𝑑𝐸 =
𝑞
𝑞
𝑑𝑞′
𝑘 2 𝑟
𝑟
– Very far from a charge distribution, it looks like a point
charge
– So if we “chop” up the distribution into small enough
pieces, each one will have a field contribution we can
calculate
– The principle of superposition then allows the
integrand to approach the true field
Continuous Charge Distributions
• General procedure to setup the integrals
– Prepare your integral
– Change integral to integrate over where the
charge lies (aka parameterization)
– Identify elements of the integrand that depend on
the integrating variable
– Determine explicit relationships with the
integrating variable
– Integrate
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
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
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 allows it
𝑆
𝑞𝑒𝑛𝑐
𝐸 ∙ 𝑑𝐴 =
𝜀𝑜
Potential
• What does it tell me?
– The change in potential energy per unit charge an
object has when moved between two points
Δ𝑈
Δ𝑉 ≡
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
• Why do I care? (cont.)
– If you have the potential defined over a small
area, the potential function encodes the
information about the electric field in the
derivative
𝐸 = −𝛻𝑉
𝐸𝑥
𝜕𝑉
𝜕𝑉
𝜕𝑉
=−
; 𝐸𝑦 = −
; 𝐸𝑧 = −
𝜕𝑥
𝜕𝑦
𝜕𝑧
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
Capacitance
• What does it tell me?
– The charge that accumulates on two conductors is
proportional to the voltage between them
• 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
• In circuits
– In well-behaved configurations, capacitors may be
combined into a single equivalent capacitor
– Parallel
𝐶𝑒𝑞 =
𝐶𝑖
* This is like increasing the area of the plates *
– Series
𝐶𝑒𝑞 =
1
𝐶𝑖
−1
* This is like increasing the separation distance *
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
𝜅=
𝜀𝑜
Capacitance
• Capacitors are in equilibrium…
– Series: when they have the same charge
– Parallel: when they have the same voltage
Practice Problems
Practice Problem
Practice Problem
Practice Problem
Practice Problem
Practice Problem