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
Flux
Gauss’s Law
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 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?
– 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′
𝑟 − 𝑟′ 3
k: Coulomb’s Constant
• 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
• Difficulties in predicting the field due to a
continuous charge distribution:
– The distribution may have an odd shape
– The charge density may change through the
distribution
• This suggests an approach via calculus is
appropriate
Continuous Charge Distributions
• Motivation for the equation:
𝑑𝑞′
𝐸 𝑟 = 𝑑𝐸 = 𝑘
𝑟 − 𝑟′
𝑞
𝑞
3
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
– Write the general integral down
– Draw a diagram and label all the parts of the 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,
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
velocity, it looks like this:
Flux
• For curved surfaces and varying flows, if we chop the
surface up into small enough pieces so that the
surface is flat and the velocity uniform, then we can
use an integral to sum up all the little “pieces” of flux
Φ=
𝑣 ∙ 𝑑𝐴
𝑆
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
𝐸 ∙ 𝑑𝐴 =
𝜀𝑜
𝑆
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